System and method for variance-based photovoltaic fleet power statistics building with the aid of a digital computer

ABSTRACT

The calculation of the variance of a correlation coefficient matrix for a photovoltaic fleet can be completed in linear space as a function of decreasing distance between pairs of photovoltaic plant locations. When obtaining irradiance data from a satellite imagery source, irradiance statistics must first be converted from irradiance statistics for an area into irradiance statistics for an average point within a pixel in the satellite imagery. The average point statistics are then averaged across all satellite pixels to determine the average across the whole photovoltaic fleet region. Where pairs of photovoltaic systems are located too far away from each other to be statistically correlated, the correlation coefficients in the matrix for that pair of photovoltaic systems are effectively zero. Consequently, the double summation portion of the calculation can be simplified to eliminate zero values based on distance between photovoltaic plant locations, substantially decreasing the size of the problem space.

CROSS-REFERENCE TO RELATED APPLICATION

This patent application is a continuation of U.S. patent applicationSer. No. 16/267,005, filed Feb. 4, 2019, pending, which is acontinuation of U.S. Pat. No. 10,197,705, issued Feb. 5, 2019; which isa continuation of U.S. Pat. No. 9,411,073, issued Aug. 9, 2016; which isa continuation-in-part of U.S. Pat. No. 8,437,959, issued May 7, 2013;which is a continuation of U.S. Pat. No. 8,335,649, issued Dec. 18,2012; which is a continuation of U.S. Pat. No. 8,165,812, issued Apr.24, 2012, the priority dates of which are claimed and the disclosures ofwhich are incorporated by reference.

This invention was made with State of California support under AgreementNumber 722. The California Public Utilities Commission of the State ofCalifornia has certain rights to this invention.

FIELD

This application relates in general to photovoltaic power generationfleet planning and operation and, in particular, to a system and methodfor variance-based photovoltaic fleet power statistics building with theaid of a digital computer.

BACKGROUND

The manufacture and usage of photovoltaic systems has advancedsignificantly in recent years due to a continually growing demand forrenewable energy resources. The cost per watt of electricity generatedby photovoltaic systems has decreased dramatically, especially whencombined with government incentives offered to encourage photovoltaicpower generation. Photovoltaic systems are widely applicable asstandalone off-grid power systems, sources of supplemental electricity,such as for use in a building or house, and as power grid-connectedsystems. Typically, when integrated into a power grid, photovoltaicsystems are collectively operated as a fleet, although the individualsystems in the fleet may be deployed at different physical locationswithin a geographic region.

Grid connection of photovoltaic power generation fleets is a fairlyrecent development. In the United States, the Energy Policy Act of 1992deregulated power utilities and mandated the opening of access to powergrids to outsiders, including independent power providers, electricityretailers, integrated energy companies, and Independent System Operators(ISOs) and Regional Transmission Organizations (RTOs). A power grid isan electricity generation, transmission, and distribution infrastructurethat delivers electricity from supplies to consumers. As electricity isconsumed almost immediately upon production, power generation andconsumption must be balanced across the entire power grid. A large powerfailure in one part of the grid could cause electrical current toreroute from remaining power generators over transmission lines ofinsufficient capacity, which creates the possibility of cascadingfailures and widespread power outages.

As a result, planners and operators of power grids need to be able toaccurately gauge both on-going and forecasted power generation andconsumption. Photovoltaic fleets participating as part of a power gridare expected to exhibit predictable power generation behaviors. Powerproduction data is needed at all levels of a power grid to which aphotovoltaic fleet is connected. Accurate power production data isparticularly crucial when a photovoltaic fleet makes a significantcontribution to a power grid's overall energy mix. At the individualphotovoltaic plant level, power production forecasting first involvesobtaining a prediction of solar irradiance, which can be derived fromground-based measurements, satellite imagery, numerical weatherprediction models, or other sources. The predicted solar irradiance dataand each photovoltaic plant's system configuration is then combined witha photovoltaic simulation model, which generates a forecast ofindividual plant power output production. The individual photovoltaicplant forecasts can then be combined into a photovoltaic powergeneration fleet forecast, such as described in commonly-assigned U.S.Pat. Nos. 8,165,811; 8,165,812; 8,165,813, all issued to Hoff on Apr.24, 2012; U.S. Pat. Nos. 8,326,535; 8,326,536, issued to Hoff on Dec. 4,2012; and U.S. Pat. No. 8,335,649, issued to Hoff on Dec. 18, 2012, thedisclosures of which are incorporated by reference.

A grid-connected photovoltaic fleet can be operationally dispersed overa neighborhood, utility region, or several states, and its constituentphotovoltaic systems (or plants) may be concentrated together or spreadout. Regardless, the aggregate grid power contribution of a photovoltaicfleet is determined as a function of the individual power contributionsof its constituent photovoltaic plants, which, in turn, may havedifferent system configurations and power capacities. Photovoltaicsystem configurations are critical to forecasting plant power output.Inaccuracies in the assumed specifications of photovoltaic systemconfigurations directly translate to inaccuracies in their power outputforecasts. Individual photovoltaic system configurations may vary basedon power rating and electrical characteristics and by their operationalfeatures, such azimuth and tilt angles and shading or other physicalobstructions.

Photovoltaic system power output is particularly sensitive to shadingdue to cloud cover, and a photovoltaic array with only a small portioncovered in shade can suffer a dramatic decrease in power output. For asingle photovoltaic system, power capacity is measured by the maximumpower output determined under standard test conditions and is expressedin units of Watt peak (Wp). However, at any given time, the actual powercould vary from the rated system power capacity depending upongeographic location, time of day, weather conditions, and other factors.Moreover, photovoltaic fleets that combine individual plants physicallyscattered over a large geographical area may be subject to differentlocation-specific weather conditions due to cloud cover and cloud speedwith a consequential affect on aggregate fleet power output.

Consequently, photovoltaic fleets operating under cloudy conditions,particularly when geographically dispersed, can exhibit variable andpotentially unpredictable performance. Conventionally, fleet variabilityis determined by collecting and feeding direct power measurements fromindividual photovoltaic systems or equivalent indirectly-derived powermeasurements into a centralized control computer or similar arrangement.To be of optimal usefulness, the direct power measurement data must becollected in near real time at fine-grained time intervals to enable ahigh resolution time series of power output to be created. However, thepracticality of such an approach diminishes as the number of systems,variations in system configurations, and geographic dispersion of thephotovoltaic fleet grow. Moreover, the costs and feasibility ofproviding remote power measurement data can make high speed datacollection and analysis insurmountable due to the bandwidth needed totransmit and the storage space needed to contain collected measurements,and the processing resources needed to scale quantitative powermeasurement analyses upwards as the fleet size grows.

For instance, one direct approach to obtaining high speed time seriespower production data from a fleet of existing photovoltaic systems isto install physical meters on every photovoltaic system, record theelectrical power output at a desired time interval, such as every 10seconds, and sum the recorded output across all photovoltaic systems inthe fleet at each time interval. The totalized power data from thephotovoltaic fleet could then be used to calculate the time-averagedfleet power, variance of fleet power, and similar values for the rate ofchange of fleet power. An equivalent direct approach to obtaining highspeed time series power production data for a future photovoltaic fleetor an existing photovoltaic fleet with incomplete metering and telemetryis to collect solar irradiance data from a dense network of weathermonitoring stations covering all anticipated locations of interest atthe desired time interval, use a photovoltaic performance model tosimulate the high speed time series output data for each photovoltaicsystem individually, and then sum the results at each time interval.

With either direct approach to obtaining high speed time series powerproduction data, several difficulties arise. First, in terms of physicalplant, calibrating, installing, operating, and maintaining meters andweather stations is expensive and detracts from cost savings otherwiseafforded through a renewable energy source. Similarly, collecting,validating, transmitting, and storing high speed data for everyphotovoltaic system or location requires collateral data communicationsand processing infrastructure, again at possibly significant expense.Moreover, data loss occurs whenever instrumentation or datacommunications do not operate reliably.

Second, in terms of inherent limitations, both direct approaches toobtaining high speed time series power production data only work fortimes, locations, and photovoltaic system configurations when and wheremeters are pre-installed; thus, high speed time series power productiondata is unavailable for all other locations, time periods, andphotovoltaic system configurations. Both direct approaches also cannotbe used to directly forecast future photovoltaic system performancesince meters must be physically present at the time and location ofinterest. Fundamentally, data also must be recorded at the timeresolution that corresponds to the desired output time resolution. Whilelow time-resolution results can be calculated from high resolution data,the opposite calculation is not possible. For example, photovoltaicfleet behavior with a 10-second resolution can not be determined fromdata collected by existing utility meters that collect the data with a15-minute resolution.

The few solar data networks that exist in the United States, such as theARM network, described in G.M. Stokes et al., “The atmospheric radiationmeasurement (ARM) program: programmatic background and design of thecloud and radiation test bed,” Bulletin of Am. Meteor. Soc., Vol. 75,pp. 1201-1221 (1994), the disclosure of which is incorporated byreference, and the SURFRAD network, do not have high density networks(the closest pair of stations in the ARM network are 50 km apart) norhave they been collecting data at a fast rate (the fastest rate is 20seconds in the ARM network and one minute in the SURFRAD network). Thelimitations of the direct measurement approaches have promptedresearchers to evaluate other alternatives. Researchers have installeddense networks of solar monitoring devices in a few limited locations,such as described in S. Kuszamaul et al., “Lanai High-Density IrradianceSensor Network for Characterizing Solar Resource Variability of MW-ScalePV System.” 35^(th) Photovoltaic Specialists Conf., Honolulu, HI (Jun.20-25, 2010), and R. George, “Estimating Ramp Rates for Large PV SystemsUsing a Dense Array of Measured Solar Radiation Data,” Am. Solar EnergySociety Annual Conf. Procs., Raleigh, NC (May 18, 2011), the disclosuresof which are incorporated by reference. As data are being collected, theresearchers examine the data to determine if there are underlying modelsthat can translate results from these devices to photovoltaic fleetproduction at a much broader area, yet fail to provide translation ofthe data.

In addition, half-hour or hourly satellite irradiance data for specificlocations and time periods of interest have been combined with randomlyselected high speed data from a limited number of ground-based weatherstations, such as described in CAISO 2011. “Summary of PreliminaryResults of 33% Renewable Integration Study—2010,” Cal. Public Util.Comm. LTPP No. R.10-05-006 (Apr. 29, 2011) and J. Stein, “Simulation of1-Minute Power Output from Utility-Scale Photovoltaic GenerationSystems,” Am. Solar Energy Society Annual Conf. Procs., Raleigh, NC (May18, 2011), the disclosures of which are incorporated by reference. Thisapproach, however, does not produce time synchronized photovoltaic fleetvariability for any particular time period because the locations of theground-based weather stations differ from the actual locations of thefleet. While such results may be useful as input data to photovoltaicsimulation models for purpose of performing high penetrationphotovoltaic studies, they are not designed to produce data that couldbe used in grid operational tools.

Many of the concerns relating to high speed time series power productiondata acquisition also apply to photovoltaic fleet output estimation.Creating a fleet power forecast requires evaluation of the solarirradiance expected over each location within a photovoltaic fleet,which must be inferred for those locations where measurement-basedsources of historical solar irradiance data are lacking. Moreover, solarirradiance data derived from satellite imagery requires additionalprocessing prior to use in simulating fleet time series output data.Satellite imagery is considered to be a set of area values that must beconverted into point values by first determining area solar irradiancestatistics followed by finding point irradiance statistics as correlatedacross the individual plant locations within the fleet. The correlationof the point statistics, though, requires finding solutions in a problemspace that grows exponentially with the number of locations. Forexample, a fleet with 10,000 photovoltaic systems would require thecomputation of a correlation coefficient matrix with 100 millioncalculations. Accordingly, the determination of inter-fleet correlationand expected fleet power, especially in the near term, becomescomputationally impracticable as the fleet's size increases.

Therefore, a need remains for an approach to efficiently correlatingsolar irradiance statistics across a plurality of photovoltaic locationsfor use in forecasting fleet power output.

SUMMARY

The calculation of the variance of a correlation coefficient matrix fora photovoltaic fleet, such as required when using satellite or otherforms of area irradiance data, can be completed in linear space as afunction of decreasing distance between pairs of photovoltaic plantlocations. When irradiance data is obtained from a satellite imagerysource, irradiance statistics must first be converted from irradiancestatistics for an area into irradiance statistics for an average pointwithin a pixel in the satellite imagery. The average point statisticsare then averaged across all satellite pixels to determine the averageacross the whole photovoltaic fleet region. Where pairs of photovoltaicsystems are located too far away from each other to be statisticallycorrelated, the correlation coefficients in the matrix for that pair ofphotovoltaic systems are effectively equal to zero. Consequently, thedouble summation portion of the calculation can be simplified toeliminate zero values based on distance between photovoltaic plantlocations and thereby substantially decrease the size of the problemspace.

In one embodiment, a system and method for variance-based photovoltaicfleet power statistics estimation with the aid of a digital computer isprovided. A set of pixels in satellite imagery data of overhead skyclearness that have been correlated to a bounded area within ageographic region is maintained in a storage, each pixel representingcollective irradiance over a plurality of points within the boundedarea, each point suitable for operation of a photovoltaic systemincluded in a photovoltaic fleet, the photovoltaic fleet connected to apower grid. an area clearness index is determined by a computer, thecomputer including a processor and coupled to the storage, based on thecollective irradiances, as represented by the set of pixels correlatedto the bounded area; A condition is set by the computer for ending adetermination of a variance of the area clearness index. A determinationof the variance of the area clearness index is performed by the computeruntil the condition is met, including: choosing a point within thebounded area; selecting at least some of the points within the boundedarea that have not already been evaluated for pairing with the chosenpoint; pairing the chosen point with each of the selected points; andcalculating a covariance for at least some of the point pairings andadding the calculated covariance for at least some of the pairs to thevariance of the area clearness index. A power statistics for thephotovoltaic fleet is built by the computer using the variance of thearea clearness index, wherein the power grid is operated based on thepower statistics.

Some of the notable elements of this methodology non-exclusivelyinclude:

(1) Employing a fully derived statistical approach to generatinghigh-speed photovoltaic fleet production data;

(2) Using a small sample of input data sources as diverse asground-based weather stations, existing photovoltaic systems, or solardata calculated from satellite images;

(3) Producing results that are usable for any photovoltaic fleetconfiguration;

(4) Supporting any time resolution, even those time resolutions fasterthan the input data collection rate; (5) Providing results in a formthat is useful and usable by electric power grid planning and operationtools; and (6) Solving solar irradiance correlation matrices as neededfor photovoltaic fleet output estimation in a linear solution space.

Still other embodiments will become readily apparent to those skilled inthe art from the following detailed description, wherein are describedembodiments by way of illustrating the best mode contemplated. As willbe realized, other and different embodiments are possible and theembodiments' several details are capable of modifications in variousobvious respects, all without departing from their spirit and the scope.Accordingly, the drawings and detailed description are to be regarded asillustrative in nature and not as restrictive.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow diagram showing a method for correlating satelliteimagery through bounded area variance for use in photovoltaic fleetoutput estimation in accordance with one embodiment.

FIG. 2 is a block diagram showing a system for correlating satelliteimagery through bounded area variance for use in photovoltaic fleetoutput estimation in accordance with one embodiment.

FIG. 3 is a graph depicting, by way of example, ten hours of time seriesirradiance data collected from a ground-based weather station with10-second resolution.

FIG. 4 is a graph depicting, by way of example, the clearness index thatcorresponds to the irradiance data presented in FIG. 3.

FIG. 5 is a graph depicting, by way of example, the change in clearnessindex that corresponds to the clearness index presented in FIG. 4.

FIG. 6 is a graph depicting, by way of example, the irradiancestatistics that correspond to the clearness index in FIG. 4 and thechange in clearness index in FIG. 5.

FIG. 7 is a flow diagram showing a function for determining variance foruse with the method of FIG. 1.

FIG. 8 is a block diagram showing, by way of example, nine evenly-spacedpoints within a three-by-three correlation region for evaluation by thefunction of FIG. 7.

FIG. 9 is a block diagram showing, by way of example, 16 evenly-spacedpoints within a three-by-three correlation region for evaluation by thefunction of FIG. 7.

FIG. 10 is a graph depicting, by way of example, the number ofcalculations required when determining variance using three differentapproaches.

FIGS. 11A-11B are photographs showing, by way of example, the locationsof the Cordelia Junction and Napa high density weather monitoringstations.

FIGS. 12A-12B are graphs depicting, by way of example, the adjustmentfactors plotted for time intervals from 10 seconds to 300 seconds.

FIGS. 13A-13F are graphs depicting, by way of example, the measured andpredicted weighted average correlation coefficients for each pair oflocations versus distance.

FIGS. 14A-14F are graphs depicting, by way of example, the sameinformation as depicted in FIGS. 13A-13F versus temporal distance.

FIGS. 15A-15F are graphs depicting, by way of example, the predictedversus the measured variances of clearness indexes using differentreference time intervals.

FIGS. 16A-16F are graphs depicting, by way of example, the predictedversus the measured variances of change in clearness indexes usingdifferent reference time intervals.

FIGS. 17A-17F are graphs and a diagram depicting, by way of example,application of the methodology described herein to the Napa network.

FIG. 18 is a graph depicting, by way of example, an actual probabilitydistribution for a given distance between two pairs of locations, ascalculated for a 1,000 meter×1,000 meter grid in one square meterincrements.

FIG. 19 is a graph depicting, by way of example, a matching of theresulting model to an actual distribution.

FIG. 20 is a graph depicting, by way of example, results generated byapplication of Equation (65).

FIG. 21 is a graph depicting, by way of example, the probability densityfunction when regions are spaced by zero to five regions.

FIG. 22 is a graph depicting, by way of example, results by applicationof the model.

DETAILED DESCRIPTION

Photovoltaic cells employ semiconductors exhibiting a photovoltaiceffect to generate direct current electricity through conversion ofsolar irradiance. Within each photovoltaic cell, light photons exciteelectrons in the semiconductors to create a higher state of energy,which acts as a charge carrier for the electrical current. The directcurrent electricity is converted by power inverters into alternatingcurrent electricity, which is then output for use in a power grid orother destination consumer. A photovoltaic system uses one or morephotovoltaic panels that are linked into an array to convert sunlightinto electricity. A single photovoltaic plant can include one or more ofthese photovoltaic arrays. In turn, a collection of photovoltaic plantscan be collectively operated as a photovoltaic fleet that is integratedinto a power grid, although the constituent photovoltaic plants withinthe fleet may actually be deployed at different physical locationsspread out over a geographic region.

To aid with the planning and operation of photovoltaic fleets, whetherat the power grid, supplemental, or standalone power generation levels,photovoltaic power plant production and variability must be estimatedexpeditiously, particularly when forecasts are needed in the near term.FIG. 1 is a flow diagram showing a method 10 for correlating satelliteimagery through bounded area variance for use in photovoltaic fleetoutput estimation in accordance with one embodiment. The method 10 canbe implemented in software and execution of the software can beperformed on a computer system, such as further described infra, as aseries of process or method modules or steps.

A time series of solar irradiance or photovoltaic (“PV”) data is firstobtained (step 11) for a set of locations representative of thegeographic region within which the photovoltaic fleet is located orintended to operate, as further described infra with reference to FIG.3. Each time series contains solar irradiance observations measured orderived, then electronically recorded at a known sampling rate at fixedtime intervals, such as at half-hour intervals, over successiveobservational time periods. The solar irradiance observations caninclude solar irradiance measured by a representative set ofground-based weather stations (step 12), existing photovoltaic systems(step 13), satellite observations (step 14), or some combinationthereof. Other sources of the solar irradiance data are possible,including numeric weather prediction models.

Next, the solar irradiance data in the time series is converted overeach of the time periods, such as at half-hour intervals, into a set ofglobal horizontal irradiance clear sky indexes, which are calculatedrelative to clear sky global horizontal irradiance based on the type ofsolar irradiance data, such as described in commonly-assigned U.S. Pat.No. 10,409,925, issued Sep. 10, 2019, the disclosure of which isincorporated by reference. The set of clearness indexes are interpretedinto as irradiance statistics (step 15), as further described infra withreference to FIG. 4-6, and power statistics, including a time series ofthe power statistics for the photovoltaic plant, are generated (step 17)as a function of the irradiance statistics and photovoltaic plantconfiguration (step 16). The photovoltaic plant configuration includespower generation and location information, including direct current(“DC”) plant and photovoltaic panel ratings; number of power inverters;latitude, longitude and elevation; sampling and recording rates; sensortype, orientation, and number; voltage at point of delivery; trackingmode (fixed, single-axis tracking, dual-axis tracking), azimuth angle,tilt angle, row-to-row spacing, tracking rotation limit, and shading orother physical obstructions. In a further embodiment, photovoltaic plantconfiguration specifications can be inferred, which can be used tocorrect, replace or, if configuration data is unavailable, stand-in forthe plant's specifications, such as described in commonly-assigned U.S.Pat. No. 8,682,585, the disclosure of which is incorporated byreference. Other types of information can also be included as part ofthe photovoltaic plant configuration. The resultant high-speed timeseries plant performance data can be combined to estimate photovoltaicfleet power output and variability, such as described incommonly-assigned U.S. Pat. Nos. 8,165,811; 8,165,812; 8,165,813;8,326,535; 8,335,649; and 8,326,536, cited supra, for use by power gridplanners, operators and other interested parties.

The calculated irradiance statistics are combined with the photovoltaicfleet configuration to generate the high-speed time series photovoltaicproduction data. In a further embodiment, the foregoing methodology mayalso require conversion of weather data for a region, such as data fromsatellite regions, to average point weather data. A non-optimizedapproach would be to calculate a correlation coefficient matrixon-the-fly for each satellite data point. Alternatively, a conversionfactor for performing area-to-point conversion of satellite imagery datais described in commonly-assigned U.S. Pat. Nos. 8,165,813 and8,326,536, cited supra.

Each forecast of power production data for a photovoltaic plant predictsthe expected power output over a forecast period. FIG. 2 is a blockdiagram showing a system 20 for correlating satellite imagery throughbounded area variance for use in photovoltaic fleet output estimation inaccordance with one embodiment. Time series power output data 32 for aphotovoltaic plant is generated using observed field conditions relatingto overhead sky clearness. Solar irradiance 23 relative to prevailingcloudy conditions 22 in a geographic region of interest is measured.Direct solar irradiance measurements can be collected by ground-basedweather stations 24. Solar irradiance measurements can also be derivedor inferred by the actual power output of existing photovoltaic systems25. Additionally, satellite observations 26 can be obtained for thegeographic region. In a further embodiment, the solar irradiance can begenerated by numerical weather prediction models. Both the direct andinferred solar irradiance measurements are considered to be sets ofpoint values that relate to a specific physical location, whereassatellite imagery data is considered to be a set of area values thatneed to be converted into point values, such as described incommonly-assigned U.S. Pat. Nos. 8,165,813 and 8,326,536, cited supra.Still other sources of solar irradiance measurements are possible.

The solar irradiance measurements are centrally collected by a computersystem 21 or equivalent computational device. The computer system 21executes the methodology described supra with reference to FIG. 1 and asfurther detailed herein to generate time series power data 26 and otheranalytics, which can be stored or provided 27 to planners, operators,and other parties for use in solar power generation 28 planning andoperations. In a further embodiment, the computer system 21 executes themethodology described infra beginning with reference to FIG. 19 forinferring operational specifications of a photovoltaic power generationsystem, which can be stored or provided 27 to planners and otherinterested parties for use in predicting individual and fleet poweroutput generation. The data feeds 29 a-c from the various sources ofsolar irradiance data need not be high speed connections; rather, thesolar irradiance measurements can be obtained at an input datacollection rate and application of the methodology described hereinprovides the generation of an output time series at any time resolution,even faster than the input time resolution. The computer system 21includes hardware components conventionally found in a general purposeprogrammable computing device, such as a central processing unit,memory, user interfacing means, such as a keyboard, mouse, and display,input/output ports, network interface, and non-volatile storage, andexecute software programs structured into routines, functions, andmodules for execution on the various systems. In addition, otherconfigurations of computational resources, whether provided as adedicated system or arranged in client-server or peer-to-peertopologies, and including unitary or distributed processing,communications, storage, and user interfacing, are possible.

The detailed steps performed as part of the methodology described suprawith reference to FIG. 1 will now be described.

Obtain Time Series Irradiance Data

The first step is to obtain time series irradiance data fromrepresentative locations. This data can be obtained from ground-basedweather stations, existing photovoltaic systems, a satellite network, orsome combination sources, as well as from other sources. The solarirradiance data is collected from several sample locations across thegeographic region that encompasses the photovoltaic fleet.

Direct irradiance data can be obtained by collecting weather data fromground-based monitoring systems. FIG. 3 is a graph depicting, by way ofexample, ten hours of time series irradiance data collected from aground-based weather station with 10-second resolution, that is, thetime interval equals ten seconds. In the graph, the blue line 32 is themeasured horizontal irradiance and the black line 31 is the calculatedclear sky horizontal irradiance for the location of the weather station.

Irradiance data can also be inferred from select photovoltaic systemsusing their electrical power output measurements. A performance modelfor each photovoltaic system is first identified, and the input solarirradiance corresponding to the power output is determined.

Finally, satellite-based irradiance data can also be used. As satelliteimagery data is pixel-based, the data for the geographic region isprovided as a set of pixels, which span across the region andencompassing the photovoltaic fleet.

Calculate Irradiance Statistics

The time series irradiance data for each location is then converted intotime series clearness index data, which is then used to calculateirradiance statistics, as described infra.

Clearness Index (Kt)

The clearness index (Kt) is calculated for each observation in the dataset. In the case of an irradiance data set, the clearness index isdetermined by dividing the measured global horizontal irradiance by theclear sky global horizontal irradiance, may be obtained from any of avariety of analytical methods. FIG. 4 is a graph depicting, by way ofexample, the clearness index that corresponds to the irradiance datapresented in FIG. 3. Calculation of the clearness index as describedherein is also generally applicable to other expressions of irradianceand cloudy conditions, including global horizontal and direct normalirradiance.

Change in Clearness Index (ΔKt)

The change in clearness index (ΔKt) over a time increment of Δt is thedifference between the clearness index starting at the beginning of atime increment t and the clearness index starting at the beginning of atime increment t, plus a time increment Δt. FIG. 5 is a graph depicting,by way of example, the change in clearness index that corresponds to theclearness index presented in FIG. 4.

Time Period

The time series data set is next divided into time periods, forinstance, from five to sixty minutes, over which statisticalcalculations are performed. The determination of time period is selecteddepending upon the end use of the power output data and the timeresolution of the input data. For example, if fleet variabilitystatistics are to be used to schedule regulation reserves on a 30-minutebasis, the time period could be selected as 30 minutes. The time periodmust be long enough to contain a sufficient number of sampleobservations, as defined by the data time interval, yet be short enoughto be usable in the application of interest. An empirical investigationmay be required to determine the optimal time period as appropriate.

Fundamental Statistics

Table 1 lists the irradiance statistics calculated from time series datafor each time period at each location in the geographic region. Notethat time period and location subscripts are not included for eachstatistic for purposes of notational simplicity.

TABLE 1 Statistic Variable Mean clearness index μ_(Kt) Varianceclearness index σ_(Kt) ² Mean clearness index change μ_(ΔKt) Varianceclearness index change σ_(ΔKt) ²

Table 2 lists sample clearness index time series data and associatedirradiance statistics over five-minute time periods. The data is basedon time series clearness index data that has a one-minute time interval.The analysis was performed over a five-minute time period. Note that theclearness index at 12:06 is only used to calculate the clearness indexchange and not to calculate the irradiance statistics.

TABLE 2 Clearness Clearness Index Index (Kt) Change (ΔKt) 12:00 50% 40%12:01 90%  0% 12:02 90% −80%  12:03 10%  0% 12:04 10% 80% 12:05 90%−40%  12:06 50% Mean (μ) 57%  0% Variance (σ²) 13% 27%

The mean clearness index change equals the first clearness index in thesucceeding time period, minus the first clearness index in the currenttime period divided by the number of time intervals in the time period.The mean clearness index change equals zero when these two values arethe same. The mean is small when there are a sufficient number of timeintervals. Furthermore, the mean is small relative to the clearnessindex change variance. To simplify the analysis, the mean clearnessindex change is assumed to equal zero for all time periods.

FIG. 6 is a graph depicting, by way of example, the irradiancestatistics that correspond to the clearness index in FIG. 4 and thechange in clearness index in FIG. 5 using a half-hour hour time period.Note that FIG. 6 presents the standard deviations, determined as thesquare root of the variance, rather than the variances, to present thestandard deviations in terms that are comparable to the mean.

Calculate Fleet Irradiance Statistics

Irradiance statistics were calculated in the previous section for thedata stream at each sample location in the geographic region. Themeaning of these statistics, however, depends upon the data source.Irradiance statistics calculated from ground-based weather station datarepresent results for a specific geographical location as pointstatistics. Irradiance statistics calculated from satellite datarepresent results for a region as area statistics. For example, if asatellite pixel corresponds to a one square kilometer grid, then theresults represent the irradiance statistics across a physical area onekilometer square.

Average irradiance statistics across the photovoltaic fleet region are acritical part of the methodology described herein. This section presentsthe steps to combine the statistical results for individual locationsand calculate average irradiance statistics for the region as a whole.The steps differ depending upon whether point statistics or areastatistics are used.

Irradiance statistics derived from ground-based sources simply need tobe averaged to form the average irradiance statistics across thephotovoltaic fleet region. Irradiance statistics from satellite sourcesare first converted from irradiance statistics for an area intoirradiance statistics for an average point within the pixel. The averagepoint statistics are then averaged across all satellite pixels todetermine the average across the photovoltaic fleet region.

Mean Clearness Index (μ _(Kt) ) and Mean Change in Clearness Index (μ_(ΔKt) )

The mean clearness index should be averaged no matter what input datasource is used, whether ground, satellite, or photovoltaic systemoriginated data. If there are N locations, then the average clearnessindex across the photovoltaic fleet region is calculated as follows.

$\begin{matrix}{\mu_{\overset{\_}{Kt}} = {\sum\limits_{i = 1}^{N}\frac{\mu_{{Kt}_{i}}}{N}}} & (1)\end{matrix}$

The mean change in clearness index for any period is assumed to be zero.As a result, the mean change in clearness index for the region is alsozero.

μ _(ΔKt) =0   (2)

Convert Area Variance to Point Variance

The following calculations are required if satellite data is used as thesource of irradiance data. Satellite observations represent valuesaveraged across the area of the pixel, rather than single pointobservations. The clearness index derived from this data (Kt^(Area)) maytherefore be considered an average of many individual pointmeasurements.

$\begin{matrix}{{Kt^{Area}} = {\sum\limits_{i = 1}^{N}\frac{Kt^{i}}{N}}} & (3)\end{matrix}$

As a result, the variance of the area clearness index based on satellitedata can be expressed as the variance of the average clearness indexesacross all locations within the satellite pixel.

$\begin{matrix}{\sigma_{{Kt} - {Area}}^{2} = {{{VAR}\left\lbrack {Kt^{Area}} \right\rbrack} = {{VAR}\left\lbrack {\sum\limits_{i = 1}^{N}\frac{Kt^{j}}{N}} \right\rbrack}}} & (4)\end{matrix}$

The variance of a sum, however, equals the sum of the covariance matrix.

$\begin{matrix}{\sigma_{{Kt} - {Area}}^{2} = {\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{{COV}\left\lbrack {{Kt^{j}},{Kt}^{j}} \right\rbrack}}}}} & (5)\end{matrix}$

Let ρ^(Kt) ^(i) ^(,Kt) ^(j) represents the correlation coefficientbetween the clearness index at location i and location j within thesatellite pixel. By definition of correlation coefficient, COV[Kt^(i),Kt^(j)]=σ^(i) _(Kt)σ^(j) _(Kt)ρ^(Kt) ^(i) ^(,Kt) ^(j) .Furthermore, since the objective is to determine the average pointvariance across the satellite pixel, the standard deviation at any pointwithin the satellite pixel can be assumed to be the same and equalsσ_(Kt), which means that σ_(Kt) ^(i)σ_(Kt) ^(j)=σ_(Kt) ² for alllocation pairs. As a result, COV [Kt^(i),Kt^(j)]=σ² _(Kt)ρ^(Kt) ^(i)^(,Kt) ^(j) . Substituting this result into Equation (5) and simplify.

$\begin{matrix}{\sigma_{{Kt} - {Area}}^{2} = {{\sigma_{Kt}^{2}\left( \frac{1}{N^{2}} \right)}{\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\rho^{{Kt^{i}},{Kt^{j}}}}}}} & (6)\end{matrix}$

Suppose that data was available to calculate the correlation coefficientin Equation (6). The computational effort required to perform a doublesummation for many points can be quite large and computationallyresource intensive. For example, a satellite pixel representing a onesquare kilometer area contains one million square meter increments. Withone million increments, Equation (6) would require one trillioncalculations to compute.

The calculation can be simplified by conversion into a continuousprobability density function of distances between location pairs acrossthe pixel and the correlation coefficient for that given distance, asfurther described infra. Thus, the irradiance statistics for a specificsatellite pixel, that is, an area statistic, rather than a pointstatistic, can be converted into the irradiance statistics at an averagepoint within that pixel by dividing by a “Area” term (A), whichcorresponds to the area of the satellite pixel. Furthermore, theprobability density function and correlation coefficient functions aregenerally assumed to be the same for all pixels within the fleet region,making the value of A constant for all pixels and reducing thecomputational burden further. Details as to how to calculate A are alsofurther described infra.

$\begin{matrix}{{\sigma_{Kt}^{2} = \frac{\sigma_{{Kt} - {Area}}^{2}}{A_{Kt}^{{Satellite}\mspace{14mu} {Pixel}}}}{{where}\text{:}}} & (7) \\{A_{Kt}^{{Satellite}\mspace{14mu} {Pixel}} = {\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\rho^{i,j}}}}} & (8)\end{matrix}$

Likewise, the change in clearness index variance across the satelliteregion can also be converted to an average point estimate using asimilar conversion factor, A_(ΔKt) ^(Area).

$\begin{matrix}{\sigma_{\Delta Kt}^{2} = \frac{\sigma_{{\Delta \; {Kt}} - {Area}}^{2}}{A_{\Delta \; {Kt}}^{{Satellite}\mspace{14mu} {Pixel}}}} & (9)\end{matrix}$

Variance of Clearness Index

$\left( {\sigma \frac{2}{Kt}} \right)$

and Variance or Change in Clearness Index

$\left( {\sigma \frac{2}{\Delta \; {Kt}}} \right)$

At this point, the point statistics (σ_(Kt) ² and σ_(ΔKt) ²) have beendetermined for each of several representative locations within the fleetregion. These values may have been obtained from either ground-basedpoint data or by converting satellite data from area into pointstatistics. If the fleet region is small, the variances calculated ateach location i can be averaged to determine the average point varianceacross the fleet region. If there are N locations, then average varianceof the clearness index across the photovoltaic fleet region iscalculated as follows.

$\begin{matrix}{{\sigma \frac{2}{Kt}} = {\sum\limits_{i = 1}^{N}\frac{\sigma_{Kt_{i}}^{2}}{N}}} & (10)\end{matrix}$

Likewise, the variance of the clearness index change is calculated asfollows.

$\begin{matrix}{{\sigma \frac{2}{\Delta \; {Kt}}} = {\sum\limits_{i = 1}^{N}\frac{\sigma_{\Delta \; {Kt}_{i}}^{2}}{N}}} & (11)\end{matrix}$

Calculate Fleet Power Statistics

The next step is to calculate photovoltaic fleet power statistics usingthe fleet irradiance statistics, as determined supra, and physicalphotovoltaic fleet configuration data. These fleet power statistics arederived from the irradiance statistics and have the same time period.

The critical photovoltaic fleet performance statistics that are ofinterest are the mean fleet power, the variance of the fleet power, andthe variance of the change in fleet power over the desired time period.As in the case of irradiance statistics, the mean change in fleet poweris assumed to be zero.

Photovoltaic System Power for Single System at Time t

Photovoltaic system power output (kW) is approximately linearly relatedto the AC-rating of the photovoltaic system (R in units of kW_(AC))times plane-of-array irradiance. Plane-of-array irradiance can berepresented by the clearness index over the photovoltaic system (KtPV)times the clear sky global horizontal irradiance times an orientationfactor (O), which both converts global horizontal irradiance toplane-of-array irradiance and has an embedded factor that convertsirradiance from Watts/m² to kW output/kW of rating. Thus, at a specificpoint in time (t), the power output for a single photovoltaic system (n)equals:

P_(t) ^(n)=R^(n)O_(t) ^(n)KtPV_(t) ^(n)I_(t) ^(Clear,n)   (12)

The change in power equals the difference in power at two differentpoints in time.

ΔP _(t,Δt) ^(n) =R ^(n) O _(t+Δt) ^(n) KtPV_(t+Δt) ^(n) I _(t+Δt)^(Clear,n) −R ^(n) O _(t) ^(n) KtPV_(t) ^(n) I _(t) ^(Clear,n)   (13)

The rating is constant, and over a short time interval, the two clearsky plane-of-array irradiances are approximately the same (O_(t+Δt)^(n)I_(t+Δt) ^(Clear,n)≈O_(t) ^(n) I _(t) ^(Clear,n)), so that the threeterms can be factored out and the change in the clearness index remains.

ΔP_(t,Δt) ^(n)≈R^(n)O_(t) ^(n) I _(t) ^(Clear,n)ΔKtPV_(t) ^(n)   (14)

Time Series Photovoltaic Power for Single System

P^(n) is a random variable that summarizes the power for a singlephotovoltaic system n over a set of times for a given time interval andset of time periods. ΔP^(n) is a random variable that summarizes thechange in power over the same set of times.

Mean Fleet Power (μ_(P))

The mean power for the fleet of photovoltaic systems over the timeperiod equals the expected value of the sum of the power output from allof the photovoltaic systems in the fleet.

$\begin{matrix}{\mu_{P} = {E\left\lbrack {\sum\limits_{n = 1}^{N}{R^{n}O^{n}KtPV^{n}I^{{Clear},n}}} \right\rbrack}} & (15)\end{matrix}$

If the time period is short and the region small, the clear skyirradiance does not change much and can be factored out of theexpectation.

$\begin{matrix}{\mu_{P} = {\mu_{I^{Clear}}{E\left\lbrack {\sum\limits_{n = 1}^{N}{R^{n}O^{n}KtPV^{n}}} \right\rbrack}}} & (16)\end{matrix}$

Again, if the time period is short and the region small, the clearnessindex can be averaged across the photovoltaic fleet region and any givenorientation factor can be assumed to be a constant within the timeperiod. The result is that:

μ_(P) =R ^(Adj.Fleet) μ_(I) _(Clear) μ _(Kt)   (17)

where μ_(I) _(Clear) is calculated, μ _(Kt) is taken from Equation (1)and:

$\begin{matrix}{R^{{Adj}.{Fleet}} = {\sum\limits_{n = 1}^{N}{R^{n}O^{n}}}} & (18)\end{matrix}$

This value can also be expressed as the average power during clear skyconditions times the average clearness index across the region.

μ_(P)=μ_(P) _(Clear) μ _(Kt)   (19)

Variance of Fleet Power (σ_(P) ²)

The variance of the power from the photovoltaic fleet equals:

$\begin{matrix}{\sigma_{P}^{2} = {{VAR}\left\lbrack {\sum\limits_{n = 1}^{N}{R^{n}O^{n}{KtPV}^{n}I^{{Clear},n}}} \right\rbrack}} & (20)\end{matrix}$

If the clear sky irradiance is the same for all systems, which will bethe case when the region is small and the time period is short, then:

$\begin{matrix}{\sigma_{P}^{2} = {{VAR}\left\lbrack {I^{Clear}{\sum\limits_{n = 1}^{N}{R^{n}O^{n}{KtPV}^{n}}}} \right\rbrack}} & (21)\end{matrix}$

The variance of a product of two independent random variables X, Y, thatis, VAR[XY]) equals E[X]²VAR[Y]+E[Y]²VAR[X]+VAR[X]VAR[Y]. If the Xrandom variable has a large mean and small variance relative to theother terms, then VAR[XY]≈E[X]²VAR[Y]. Thus, the clear sky irradiancecan be factored out of Equation (21) and can be written as:

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}{{VAR}\left\lbrack {\sum\limits_{n = 1}^{N}{R^{n}{KtPV}^{n}O^{n}}} \right\rbrack}}} & (22)\end{matrix}$

The variance of a sum equals the sum of the covariance matrix.

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}\left( {\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{{COV}\left\lbrack {{R^{i}{KtPV}^{i}O^{i}},{R^{j}{KtPV}^{j}O^{j}}} \right\rbrack}}} \right)}} & (23)\end{matrix}$

In addition, over a short time period, the factor to convert from clearsky GHI to clear sky POA does not vary much and becomes a constant. Allfour variables can be factored out of the covariance equation.

$\begin{matrix}{\sigma_{P}^{2} = {\left( \mu_{I^{Clear}} \right)^{2}\left( {\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{\left( {R^{i}O^{i}} \right)\left( {R^{j}O^{j}} \right){{COV}\left\lbrack {{KtPV}^{i},{KtPV}^{j}} \right\rbrack}}}} \right)}} & (24)\end{matrix}$

For any i and j,

$\begin{matrix}{\mspace{79mu} {{{COV}\left\lbrack {{KtPV}^{i},{KtPV}^{j}} \right\rbrack} = {{\sqrt{\sigma_{{KtPV}^{i}}^{2}\sigma_{{KtPV}^{j}}^{2}}{\rho^{{Kt}^{i},{Kt}^{j}}.\sigma_{P}^{2}}} = {\left( \mu_{I^{Clear}} \right)^{2}\left( {\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{\left( {R^{i}O^{i}} \right)\left( {R^{j}O^{j}} \right)\sqrt{\sigma_{{KtPV}^{i}}^{2}\sigma_{{KtPV}^{j}}^{2}}\rho^{{Kt}^{i},{Kt}^{j}}}}} \right)}}}} & (25)\end{matrix}$

As discussed supra, the variance of the satellite data required aconversion from the satellite area, that is, the area covered by apixel, to an average point within the satellite area. In the same way,assuming a uniform clearness index across the region of the photovoltaicplant, the variance of the clearness index across a region the size ofthe photovoltaic plant within the fleet also needs to be adjusted. Thesame approach that was used to adjust the satellite clearness index canbe used to adjust the photovoltaic clearness index. Thus, each varianceneeds to be adjusted to reflect the area that the i^(th) photovoltaicplant covers.

$\begin{matrix}{\sigma_{{KtPV}^{i}}^{2} = {A_{Kt}^{i}\sigma \frac{2}{Kt}}} & (26)\end{matrix}$

Substituting and then factoring the clearness index variance given theassumption that the average variance is constant across the regionyields:

$\begin{matrix}{\sigma_{P}^{2} = {\left( {R^{{Adj}.{Fleet}}\mu_{I^{Clear}}} \right)^{2}P^{Kt}\sigma \frac{2}{Kt}}} & (27)\end{matrix}$

where the correlation matrix equals:

$\begin{matrix}{P^{Kt} = \frac{\sum_{i = 1}^{N}{\sum_{j = 1}^{N}{\left( {R^{i}O^{i}A_{Kt}^{i}} \right)\left( {R^{j}O^{j}A_{Kt}^{j}} \right)\rho^{{Kt}^{i},{Kt}^{j}}}}}{\left( {\sum_{n = 1}^{N}{R^{n}O^{n}}} \right)^{2}}} & (28)\end{matrix}$

R^(Adj.Fleet) μ_(I) _(Clear) in Equation (27) can be written as thepower produced by the photovoltaic fleet under clear sky conditions,that is:

$\begin{matrix}{\sigma_{P}^{2} = {\mu_{P^{{Clear}^{2}}}P^{Kt}\sigma \frac{2}{Kt}}} & (29)\end{matrix}$

If the region is large and the clearness index mean or variances varysubstantially across the region, then the simplifications may not beable to be applied. Notwithstanding, if the simplification isinapplicable, the systems are likely located far enough away from eachother, so as to be independent. In that case, the correlationcoefficients between plants in different regions would be zero, so mostof the terms in the summation are also zero and an inter-regionalsimplification can be made. The variance and mean then become theweighted average values based on regional photovoltaic capacity andorientation.

Discussion

In Equation (28), the correlation matrix term embeds the effect ofintra-plant and inter-plant geographic diversification. The area-relatedterms (A) inside the summations reflect the intra-plant power smoothingthat takes place in a large plant and may be calculated using thesimplified relationship, as further discussed supra. These terms arethen weighted by the effective plant output at the time, that is, therating adjusted for orientation. The multiplication of these terms withthe correlation coefficients reflects the inter-plant smoothing due tothe separation of photovoltaic systems from one another.

Variance of Change in Fleet Power (σ_(ΔP) ²)

A similar approach can be used to show that the variance of the changein power equals:

$\begin{matrix}{{\sigma_{\Delta \; P}^{2} = {\mu_{P^{{Clear}^{2}}}P^{\Delta \; {Kt}}\sigma \frac{2}{\Delta \; {Kt}}}}{{where}\text{:}}} & (30) \\{P^{\Delta \; {Kt}} = \frac{\sum_{i = 1}^{N}{\sum_{j = 1}^{N}{\left( {R^{i}O^{i}A_{\Delta \; {Kt}}^{i}} \right)\left( {R^{j}O^{j}A_{\Delta \; {Kt}}^{j}} \right)\rho^{{\Delta \; {Kt}^{i}},{\Delta \; {Kt}^{j}}}}}}{\left( {\sum_{n = 1}^{N}{R^{n}O^{n}}} \right)^{2}}} & (31)\end{matrix}$

Variance of Change in Fleet Power as Decreasing Function of Distance Thedeterminations of Equations (5), (6), (8), (23), (24), (28), (30), and(31) each require solving a covariance matrix or some derivative of thecovariance matrix, which becomes increasingly computationally intensiveat an exponential or near-exponential rate as the network of pointsrepresenting each system within the photovoltaic fleet grows. Forexample, a network with 10,000 photovoltaic systems would require thecomputation of a correlation coefficient matrix with 100 millioncalculations. As well, in some applications, the computation must becompleted frequently and possibly over a near-term time window toprovide fleet-wide power output forecasting to planners and operators.The computing of 100 million covariance solutions, though, can requirean inordinate amount of computational resource expenditure in terms ofprocessing cycles, network bandwidth and storage, plus significant timefor completion, making frequent calculation effectively impracticable.

To illustrate the computational burden associated with solving acovariance matrix, consider an example of calculating the variability ofthe power output of a fleet of N photovoltaic systems. Let the poweroutput associated with each individual photovoltaic system i berepresented by the random variable P^(i). The variance of the fleetFleetVariance is defined as follows:

$\begin{matrix}{{FleetVariance}{= {{VAR}\left\lbrack {\sum\limits_{i = 1}^{N}P^{I}} \right\rbrack}}} & (i)\end{matrix}$

In general, the variance of the sum of random variables can be expressedas the sum of the covariance matrix, provided that the variables arecorrelated. Thus, Equation (i) can be rewritten as:

$\begin{matrix}{{{VAR}\left\lbrack {\sum\limits_{i = 1}^{N}P^{i}} \right\rbrack} = {\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{{COV}\left\lbrack {P^{i},P^{j}} \right\rbrack}}}} & ({ii})\end{matrix}$

Solving the covariance matrix in Equation (ii) requires N² calculations,which represents an exponential computational burden. As asimplification, the computational burden can be reduced first, byrecognizing that the covariance for a location and itself merely equalsthe variance of the location, and second, by noting that the covariancecalculation is order independent, that is, the covariance betweenlocations A and B equals the covariance between locations B and A. Thus,Equation (ii) can be rewritten as:

$\begin{matrix}{{{VAR}\left\lbrack {\sum\limits_{i = 1}^{N}P^{i}} \right\rbrack} = {{\sum\limits_{i = 1}^{N}\sigma_{P^{i}}^{2}} + {2{\sum\limits_{i = 1}^{N - 1}{\sum\limits_{j = {i + 1}}^{N}{{COV}\left\lbrack {P^{i},P^{j}} \right\rbrack}}}}}} & ({iii})\end{matrix}$

Equation (iii) reduces the number of calculations by slightly less thanhalf from N² to N[(N−1)/2+1]. This approach, however, is stillexponentially related to the number of locations in the network and cantherefore become intractable.

Another way to express each term in the covariance matrix is using thestandard deviations and Pearson's correlation coefficient. The Pearson'scorrelation coefficient between two variables is defined as thecovariance of the two variables divided by the product of their standarddeviations. Thus, Equation (iii) can be rewritten as:

$\begin{matrix}{{{VAR}\left\lbrack {\sum\limits_{i = 1}^{N}P^{i}} \right\rbrack} = {{\sum\limits_{i = 1}^{N}\sigma_{P^{i}}^{2}} + {2{\sum\limits_{i = 1}^{N - 1}{\sum\limits_{j = {i + 1}}^{N}{\rho^{P^{i},P^{j}}\sigma_{P^{i}}\sigma_{P^{j}}}}}}}} & ({iv})\end{matrix}$

However, Equation (iv) still exhibits the same computational complexityas Equation (iii) and does not substantially alleviate the computationalload.

Skipping ahead, FIGS. 13A-13F are graphs depicting, by way of example,the measured and predicted weighted average correlation coefficients foreach pair of locations versus distance. FIGS. 14A-14F are graphsdepicting, by way of example, the same information as depicted in FIGS.13A-13F versus temporal distance, based on the assumption that cloudspeed was six meters per second. The upper line and dots appearing inclose proximity to the upper line present the clearness index and thelower line and dots appearing in close proximity to the lower linepresent the change in clearness index for time intervals from 10 secondsto 5 minutes. The symbols are the measured results and the lines are thepredicted results. FIGS. 13A-13F and 14A-14F suggest that thecorrelation coefficient for sky clearness between two locations, whichis the critical variable in determining photovoltaic power production,is a decreasing function of the distance between the two locations. Thecorrelation coefficient is equal to 1.0 when the two systems are locatednext to each other, while the correlation coefficient approaches zero asthe distance increases, which implies that the covariance between thetwo locations beyond some distance is equal to zero, that is, the twolocations are not correlated.

The computational burden can be reduced in at least two ways. First,where pairs of photovoltaic systems are located too far away from eachother to be statistically correlated, the correlation coefficients inthe matrix for that pair of photovoltaic systems are effectively equalto zero. Consequently, the double summation portion of the variancecalculation can be simplified to eliminate zero values based on distancebetween photovoltaic plant locations. Second, once the simplificationhas been made, rather than calculating the entire covariance matrixon-the-fly for every time period, the matrix can be calculated once atthe beginning of the analysis for a variety of cloud speed conditions,after which subsequent analyses would simply perform a lookup of theappropriate pre-calculated covariance value. The zero valuesimplification of the correlation coefficient matrix will now bediscussed in detail.

As a heuristical simplification, the variance of change in fleet powercan be computed as a function of decreasing distance. When irradiancedata is obtained from a satellite imagery source, irradiance statisticsmust first be converted from irradiance statistics for an area intoirradiance statistics for an average point within a pixel in thesatellite imagery. The average point statistics are then averaged acrossall satellite pixels to determine the average across the wholephotovoltaic fleet region. FIG. 7 is a flow diagram showing a function40 for determining variance for use with the method 10 of FIG. 1. Thefunction 40 calculates the variance of the average point statisticsacross all satellite pixels for the bounded area upon which aphotovoltaic fleet either is (or can be) situated. Each point within thebounded area represents the location of a photovoltaic plant that ispart of the fleet.

The function 40 is described as a recursive procedure, but could also beequivalently expressed using an iterative loop or other form ofinstruction sequencing that allows values to be sequentially evaluated.Additionally, for convenience, an identification number from an orderedlist can be assigned to each point, and each point is then sequentiallyselected, processed and removed from the ordered list upon completion ofevaluation. Alternatively, each point can be selected such that onlythose points that have not yet been evaluated are picked (step 41). Arecursive exit condition is defined, such that if all points have beenevaluated (step 42), the function immediately exits.

Otherwise, each point is evaluated (step 42), as follows. First, a zerocorrelation value is determined (step 43). The zero correlation value isa value for the correlation coefficients of a covariance matrix, suchthat, the correlation coefficients for a given pair of points that areequal to or less than (or, alternatively, greater than) the zerocorrelation value can be omitted from the matrix with essentially noaffect on the total covariance. The zero correlation value can be usedas a form of filter on correlation coefficients in at least two ways.The zero correlation value can be set to a near-zero value.

In one embodiment where the variance between the points within a boundedarea is calculated as a bounded area variance using a continuousprobability density function, the zero correlation value is a thresholdvalue against which correlation coefficients are compared. In a furtherembodiment where the variance between the points within a bounded areais calculated as an average point variance, a zero correlation distanceis found as a function of the zero correlation value (step 44). The zerocorrelation distance is the distance beyond which there is effectivelyno correlation between the photovoltaic system output (or the change inthe photovoltaic system output) for a given pair of points. The zerocorrelation distance can be calculated, for instance, by settingEquation (44) (for power) or Equation (45) (for the change in power), asfurther discussed infra, equal to the zero correlation value and solvingfor distance. These equations provide that the correlation is a functionof distance, cloud speed, and time interval. If cloud speed variesacross the locations, then a unique zero correlation distance can becalculated for each point. In a further embodiment, there is one zerocorrelation distance for all points if cloud speed is the same acrossall locations. In a still further embodiment, a maximum (or minimum)zero correlation distance across the locations can be determined byselecting the point with the slowest (or fastest) cloud speed. Stillother ways of obtaining the zero correlation distance are possible,including evaluation of equations that express correlation as a functionof distance (and possibly other parameters) by which the equation is setto equal the zero correlation value and solved for distance.

The chosen point is paired with each of the other points that have notalready been evaluated and, if applicable, that are located within thezero correlation distance of the chosen point (step 45). In oneembodiment, the set of points located within the zero correlationdistance can be logically represented as falling within a grid orcorrelation region centered on the chosen point. The zero correlationdistance is first converted (from meters) into degrees and the latitudeand longitude coordinates for the chosen point are found. Thecorrelation region is then defined as a rectangular or square region,such that Longitude−Zero Correlation Degrees≤Longitude≤Longitude+ZeroCorrelation Degrees and Latitude−Zero CorrelationDegrees≤Latitude≤Latitude+Zero Correlation Degrees. In a furtherembodiment, the correlation region is constructed as a circular regioncentered around the chosen point. This option, however, requirescalculating the distance between the chosen point and all other possiblepoints.

Each of the pairings of chosen point-to-a point yet to be evaluated areiteratively processed (step 46) as follows. A correlation coefficientfor the point pairing is calculated (step 47). In one embodiment wherethe variance is calculated as a bounded area variance, the covariance isonly calculated (step 49) and added to the variance of the areaclearness index (step 50) if the correlation coefficient is greater thanthe zero correlation value (step 48). In a further embodiment where thevariance is calculated as an average point variance, the covariance issimply calculated (step 49) and added to the variance of the areaclearness index (step 50). Processing continues with the next pointpairing (step 51). Finally, the function 40 is recursively called toevaluate the next point, after which the variance is returned.

The variance heuristic determination can be illustrated through twoexamples. FIGS. 8 and 9 are block diagrams respectively showing, by wayof example, nine evenly-spaced points within a three-by-threecorrelation region and 16 evenly-spaced points within a three-by-threecorrelation region for evaluation by the function of FIG. 7. Referringfirst to FIG. 8, suppose that there are nine photovoltaic systemlocations, which are evenly spaced in a square three-by-three region.Evaluation proceeds row-wise, left-to-right, top-to-bottom, from theupper left-hand corner, and the chosen location is labeled with anidentification number. Within each evaluative step, the black squarewith white lettering represents the chosen point and the heavy blackborder represents the corresponding correlation region. The dark graysquares are the locations for which correlation is non-zero and thelight gray squares are the locations for which correlation is zero. Thenumber of correlation coefficients to be calculated equals the sum ofthe number of dark gray boxes, with 20 correlation correlationscalculated here. Note that while this example presents the results in aparticular order, the approach does not require the locations to beconsidered in any particular order. Referring next to FIG. 9, supposethat there are now 16 photovoltaic system locations, which are evenlyspaced in the same square three-by-three region. As before, evaluationproceeds row-wise, left-to-right, top-to-bottom, from the upperleft-hand corner, and the chosen location is labeled with anidentification number. However, in this example, there are 42correlation coefficient calculations.

The number of point pairing combinations to be calculated for differentnumbers of points and sizes of correlation regions can be determined asa function of the number of locations and the correlation region size.Due to edge effects, however, determination can be complicated and asimpler approach is to simply determine an upper bound on the number ofcorrelations to be calculated. Assuming equal spacing, the upper bound Ccan be determined as:

$\begin{matrix}{C = \frac{N \times \left( {{{Corre}{lation\_ region}} - 1} \right)}{2}} & (v)\end{matrix}$

where N is the number of points within the bounded area and Correlationregion is the number of equal-sized divisions of the bounded area.

The upper bound can be used to illustrate the reduction of the problemspace from exponential to linear. FIG. 10 is a graph depicting, by wayof example, the number of calculations required when determiningvariance using three different approaches. The x-axis represents thenumber of photovoltaic plant locations (points) within the boundedregion. The y-axis represents the upper bound on the number ofcorrelation calculations required, as determined using Equation (v).Assume, for instance, that a correlation region includes nine divisionsand that the number of possible points ranges up to 100. The totalpossible correlations required using correlation coefficient matrixcomputations without simplification is as high as 10,000 computations,whilst the total possible correlations required using the simplificationafforded by standard deviations and Pearson's correlation coefficient,per Equation (iv), is as high as 5,000 computations. By comparison, theheuristical of determining the variance of change in fleet power as afunction of decreasing distance approach requires a maximum of 400correlations calculations, which represents a drastic reduction incomputational burden without cognizable loss of accuracy in variance.

Time Lag Correlation Coefficient

The next step is to adjust the photovoltaic fleet power statistics fromthe input time interval to the desired output time interval. Forexample, the time series data may have been collected and stored every60 seconds. The user of the results, however, may want to havephotovoltaic fleet power statistics at a 10-second rate. This adjustmentis made using the time lag correlation coefficient.

The time lag correlation coefficient reflects the relationship betweenfleet power and that same fleet power starting one time interval (Δt)later. Specifically, the time lag correlation coefficient is defined asfollows:

$\begin{matrix}{\rho^{P,P^{\Delta t}} = \frac{{COV}\left\lbrack {P,P^{\Delta t}} \right\rbrack}{\sqrt{\sigma_{P}^{2}\sigma_{P^{\Delta t}}^{2}}}} & (32)\end{matrix}$

The assumption that the mean clearness index change equals zero impliesthat σ_(PΔt) ²=σ_(P) ². Given a non-zero variance of power, thisassumption can also be used to show that

$\begin{matrix}{\frac{{COV}\left\lceil {P,P^{\Delta t}} \right\rceil}{\sigma_{P}^{2}} = {1 - {\frac{\sigma_{\Delta P}^{2}}{2\sigma_{P}^{2}}.}}} & \;\end{matrix}$

Therefore:

$\begin{matrix}{\rho^{P,P^{\Delta \; t}} = {1 - \frac{\sigma_{\Delta P}^{2}}{2\sigma_{P}^{2}}}} & (33)\end{matrix}$

This relationship illustrates how the time lag correlation coefficientfor the time interval associated with the data collection rate iscompletely defined in terms of fleet power statistics alreadycalculated. A more detailed derivation is described infra.

Equation (33) can be stated completely in terms of the photovoltaicfleet configuration and the fleet region clearness index statistics bysubstituting Equations (29) and (30). Specifically, the time lagcorrelation coefficient can be stated entirely in terms of photovoltaicfleet configuration, the variance of the clearness index, and thevariance of the change in the clearness index associated with the timeincrement of the input data.

$\begin{matrix}{\rho^{P,P^{\Delta t}} = {1 - \frac{P^{\Delta Kt}\sigma_{\overset{\_}{\Delta \; {Kt}}}^{2}}{2P^{Kt}\overset{\_}{\sigma_{\overset{\_}{Kt}}^{2}}}}} & (34)\end{matrix}$

Generate High-Speed Time Series Photovoltaic Fleet Power

The final step is to generate high-speed time series photovoltaic fleetpower data based on irradiance statistics, photovoltaic fleetconfiguration, and the time lag correlation coefficient. This step is toconstruct time series photovoltaic fleet production from statisticalmeasures over the desired time period, for instance, at half-hour outputintervals.

A joint probability distribution function is required for this step. Thebivariate probability density function of two unit normal randomvariables (X and Y) with a correlation coefficient of ρ equals:

$\begin{matrix}{{f\left( {x,y} \right)} = {\frac{1}{2\pi \sqrt{1 - \rho^{2}}}{\exp \left\lbrack {- \frac{\left( {x^{2} + y^{2} - {2\rho \; {xy}}} \right)}{2\left( {1 - \rho^{2}} \right)}} \right\rbrack}}} & (35)\end{matrix}$

The single variable probability density function for a unit normalrandom variable X alone is

${f(x)} = {\frac{1}{\sqrt{2\pi}}{{\exp\left( {- \frac{x^{2}}{2}} \right)}.}}$

In addition, a conditional distribution for y can be calculated based ona known x by dividing the bivariate probability density function by thesingle variable probability density

$\left( {{i.e.},\ {{f\left( y \middle| x \right)} = \frac{f\left( {x,y} \right)}{f(x)}}} \right).$

Making the appropriate substitutions, the result is that the conditionaldistribution of y based on a known x equals:

$\begin{matrix}{{f\left( y \middle| x \right)} = {\frac{1}{\sqrt{2\pi}\sqrt{1 - \rho^{2}}}{\exp \left\lbrack {- \frac{\left( {y - {px}} \right)^{2}}{2\left( {1 - p^{2}} \right)}} \right\rbrack}}} & (36)\end{matrix}$

Define a random variable

$Z = \frac{Y - {\rho \; x}}{\sqrt{1 - \rho^{2}}}$

and substitute into Equation (36).The result is that the conditional probability of z given a known xequals:

$\begin{matrix}{{f\left( \left. z \right|_{X} \right)} = {\frac{1}{\sqrt{2\pi}}{\exp\left( {- \frac{z^{2}}{2}} \right)}}} & (37)\end{matrix}$

The cumulative distribution function for Z can be denoted by φ(z*),where z* represents a specific value for z. The result equals aprobability (p) that ranges between 0 (when z*=−∞) and 1 (when z*=∞).The function represents the cumulative probability that any value of zis less than z*, as determined by a computer program or value lookup.

$\begin{matrix}{p = {{\varphi \left( z^{*} \right)} = {\frac{1}{\sqrt{2\pi}}{\int_{- \infty}^{z^{*}}{{\exp\left( {- \frac{z^{2}}{2}} \right)}{dz}\int}}}}} & (38)\end{matrix}$

Rather than selecting z*, however, a probability p falling between 0 and1 can be selected and the corresponding z* that results in thisprobability found, which can be accomplished by taking the inverse ofthe cumulative distribution function.

φ⁻(p)=z*   (39)

Substituting back for z as defined above results in:

$\begin{matrix}{{\varphi^{- 1}(p)} = \frac{y - {\rho x}}{\sqrt{1 - \rho^{2}}}} & (40)\end{matrix}$

Now, let the random variables equal

${X = {{\frac{P - \mu_{P}}{\sigma_{P}}\mspace{14mu} {and}\mspace{14mu} Y} = \frac{P^{\Delta t} - \mu_{P^{\Delta t}}}{\sigma_{P^{\Delta t}}}}},$

with the correlation coefficient being the time lag correlationcoefficient between P and P^(Δt) (i.e., let ρ=ρ^(P,P) ^(Δt) ). When Δtis small, then the mean and standard deviations for P^(Δt) areapproximately equal to the mean and standard deviation for P. Thus, Ycan be restated as

$Y \approx {\frac{P^{\Delta t} - \mu_{P}}{\sigma_{P}}.}$

Add a time subscript to all of the relevant data to represent a specificpoint in time and substitute x, y, and ρ into Equation (40).

$\begin{matrix}{{\varphi^{- 1}(p)} = \frac{\left( \frac{{P^{\Delta \; t}t} - \mu_{P}}{\sigma_{P}} \right) - {\rho^{P,P^{\Delta \; t}}\left( \frac{P_{t} - \mu_{P}}{\sigma_{P}} \right)}}{\sqrt{1 - \rho^{P,P^{\Delta \; t^{2}}}}}} & (41)\end{matrix}$

The random variable P^(Δt), however, is simply the random variable Pshifted in time by a time interval of Δt. As a result, at any given timet, P^(Δt) _(t)=P_(t+Δt). Make this substitution into Equation (41) andsolve in terms of P_(t+Δt).

$\begin{matrix}{P_{t + {\Delta t}} = {{{\rho^{P,P}}^{\Delta t}P_{t}} + {\left( {1 - {\rho^{P,P}}^{\Delta t}} \right)\mu_{P}} + {\sqrt{\sigma_{P}^{2}\left( {1 - \rho^{P,P^{\Delta \; t^{2}}}} \right)}{\varphi^{- 1}(p)}}}} & (42)\end{matrix}$

At any given time, photovoltaic fleet power equals photovoltaic fleetpower under clear sky conditions times the average regional clearnessindex, that is, P_(t)=P_(t) ^(Clear)Kt_(t). In addition, over a shorttime period, μ_(p)≈P_(t) ^(Clear) μ _(Kt) and

$\sigma_{P}^{2} \approx {\left( P_{t}^{Clear} \right)^{2}P^{Kt}{\sigma_{\overset{\_}{Kt}}^{2}.}}$

Substitute these three relationships into Equation (42) and factor outphotovoltaic fleet power under clear sky conditions ((P_(t) ^(Clear)))as common to all three terms.

$\begin{matrix}{P_{t + {\Delta t}} = {P_{t}^{Clear}\begin{bmatrix}{{\rho^{P,P^{\Delta \; t}}Kt_{t}} + {\left( {1 - \rho^{P,P^{\Delta \; t}}} \right)\mu_{\overset{\_}{Kt}}} +} \\{\sqrt{P^{Kt}{\sigma_{\overset{\_}{Kt}}^{2}\left( {1 - \rho^{P,P^{\Delta \; t^{2}}}} \right)}}{\varphi^{- 1}\left( p_{t} \right)}}\end{bmatrix}}} & (43)\end{matrix}$

Equation (43) provides an iterative method to generate high-speed timeseries photovoltaic production data for a fleet of photovoltaic systems.At each time step (t+Δt), the power delivered by the fleet ofphotovoltaic systems (P_(t+Δt)) is calculated using input values fromtime step t. Thus, a time series of power outputs can be created. Theinputs include:

-   -   P_(t) ^(Clear)—photovoltaic fleet power during clear sky        conditions calculated using a photovoltaic simulation program        and clear sky irradiance.    -   Kt_(t)—average regional clearness index inferred based on P_(t)        calculated in time step t, that is,

${{Kt_{t}} = \frac{P_{t}}{P_{t}^{Clear}}}.$

-   -   μ _(Kt) —mean clearness index calculated using time series        irradiance data and Equation (1).

$\sigma_{\overset{\_}{Kt}}^{2}$

-   -   —variance of the clearness index calculated using time series        irradiance data and Equation (10).    -   ρ^(P,P) ^(Δt) —fleet configuration as reflected in the time lag        correlation coefficient calculated using Equation (34). In turn,        Equation (34), relies upon correlation coefficients from        Equations (28) and (31). A method to obtain these correlation        coefficients by empirical means is described in        commonly-assigned U.S. Pat. Nos. 8,165,811, and 8,165,813, the        disclosure of which is incorporated by reference.    -   P^(Kt)—fleet configuration as reflected in the clearness index        correlation coefficient matrix calculated using Equation (28)        where, again, the correlation coefficients may be obtained using        the empirical results as further described infra.    -   ϕ⁻¹(p_(t))—the inverse cumulative normal distribution function        based on a random variable between 0 and 1.

Derivation of Empirical Models

The previous section developed the mathematical relationships used tocalculate irradiance and power statistics for the region associated witha photovoltaic fleet. The relationships between Equations (8), (28),(31), and (34) depend upon the ability to obtain point-to-pointcorrelation coefficients. This section presents empirically-derivedmodels that can be used to determine the value of the coefficients forthis purpose.

A mobile network of 25 weather monitoring devices was deployed in a 400meter by 400 meter grid in Cordelia Junction, CA, between Nov. 6, 2010,and Nov. 15, 2010, and in a 4,000 meter by 4,000 meter grid in Napa,Calif., between Nov. 19, 2010, and Nov. 24, 2010. FIGS. 11A-11B arephotographs showing, by way of example, the locations of the CordeliaJunction and Napa high density weather monitoring stations.

An analysis was performed by examining results from Napa and CordeliaJunction using 10, 30, 60, 120 and 180 second time intervals over eachhalf-hour time period in the data set. The variance of the clearnessindex and the variance of the change in clearness index were calculatedfor each of the 25 locations for each of the two networks. In addition,the clearness index correlation coefficient and the change in clearnessindex correlation coefficient for each of the 625 possible pairs, 300 ofwhich are unique, for each of the two locations were calculated.

An empirical model is proposed as part of the methodology describedherein to estimate the correlation coefficient of the clearness indexand change in clearness index between any two points by using as inputsthe following: distance between the two points, cloud speed, and timeinterval. For the analysis, distances were measured, cloud speed wasimplied, and a time interval was selected.

The empirical models infra describe correlation coefficients between twopoints (i and j), making use of “temporal distance,” defined as thephysical distance (meters) between points i and j, divided by theregional cloud speed (meters per second) and having units of seconds.The temporal distance answers the question, “How much time is needed tospan two locations?”

Cloud speed was estimated to be six meters per second. Results indicatethat the clearness index correlation coefficient between the twolocations closely matches the estimated value as calculated using thefollowing empirical model:

ρ^(Kt) ^(i) ^(,Kt) ^(j) =exp(C ₁×TemporalDistance)^(ClearnessPower)  (44)

where TemporalDistance=Distance (meters)/CloudSpeed (meters per second),ClearnessPower=ln(C₂Δt)−9.3, such that 5≤k≤15, where the expected valueis k=9.3, Δt is the desired output time interval (seconds), C₁=10⁻³seconds⁻¹, and C₂=1 seconds⁻¹.

Results also indicate that the correlation coefficient for the change inclearness index between two locations closely matches the valuescalculated using the following empirical relationship:

ρ^(Δkt) ^(i) ^(,Kt) ^(j) =(ρ^(Kt) ^(i) ^(,Kt) ^(j) )^(ΔClearnessPower)  (45)

where ρ^(Kt) ^(i) ^(,Kt) ^(j) is calculated using Equation (44) and

${{\Delta \; {ClearnessPower}} = {1 + \frac{m}{C_{2}\Delta t}}},$

such that 100≤m≤200, where the expected value is m=140.

Empirical results also lead to the following models that may be used totranslate the variance of clearness index and the variance of change inclearness index from the measured time interval (Δt ref) to the desiredoutput time interval (Δt).

$\begin{matrix}{\sigma_{Kt_{\Delta \; t}}^{2} = {\sigma_{Kt_{\Delta \; t\mspace{14mu} {ref}}}^{2}{\exp \left\lbrack {1 - \left( \frac{\Delta t}{\Delta \; t\mspace{14mu} {ref}} \right)^{C_{3}}} \right\rbrack}}} & (46) \\{\sigma_{\Delta Kt_{\Delta \; t}}^{2} = {\sigma_{\Delta Kt_{\Delta \; t\mspace{14mu} {ref}}}^{2}\left\{ {1 - {2\left\lbrack {1 - \left( \frac{\Delta t}{\Delta \; t\mspace{14mu} {ref}} \right)^{C_{3}}} \right\rbrack}} \right\}}} & (47)\end{matrix}$

where C₃=0.1≤C₃≤0.2, where the expected value is C₃=0.15.

FIGS. 12A-12B are graphs depicting, by way of example, the adjustmentfactors plotted for time intervals from 10 seconds to 300 seconds. Forexample, if the variance is calculated at a 300-second time interval andthe user desires results at a 10-second time interval, the adjustmentfor the variance clearness index would be 1.49.

These empirical models represent a valuable means to rapidly calculatecorrelation coefficients and translate time interval withreadily-available information, which avoids the use ofcomputation-intensive calculations and high-speed streams of data frommany point sources, as would otherwise be required.

Validation

Equations (44) and (45) were validated by calculating the correlationcoefficients for every pair of locations in the Cordelia Junctionnetwork and the Napa network at half-hour time periods. The correlationcoefficients for each time period were then weighted by thecorresponding variance of that location and time period to determineweighted average correlation coefficient for each location pair. Theweighting was performed as follows:

${\overset{\_}{\rho^{{Kt}^{i},{Kt}^{j}}} = \frac{\sum\limits_{t = 1}^{T}{\sigma_{{{Kt} - i},j_{t}}^{2}\rho^{{Kt^{i}},{Kt^{j}}}t}}{\sum\limits_{t = 1}^{T}\sigma_{{{Kt} - i},j_{t}}^{2}}},{and}$$\overset{\_}{\rho^{{\Delta Kt^{i}},{\Delta \; {Kt}^{j}}}} = {\frac{\sum\limits_{t = 1}^{T}{\sigma_{{{\Delta \; {Kt}} - i},j_{t}}^{2}\rho^{{\Delta Kt^{i}},{\Delta \; {Kt}^{j}}}t}}{\sum\limits_{t = 1}^{T}\sigma_{{{\Delta \; {Kt}} - i},j_{t}}^{2}}.}$

FIGS. 13A-13F are graphs depicting, by way of example, the measured andpredicted weighted average correlation coefficients for each pair oflocations versus distance. FIGS. 14A-14F are graphs depicting, by way ofexample, the same information as depicted in FIGS. 13A-13F versustemporal distance, based on the assumption that cloud speed was sixmeters per second. Several observations can be drawn based on theinformation provided by the FIGS. 13A-13F and 14A-14F. First, for agiven time interval, the correlation coefficients for both the clearnessindex and the change in the clearness index follow an exponentialdecline pattern versus distance (and temporal distance). Second, thepredicted results are a good representation of the measured results forboth the correlation coefficients and the variances, even though theresults are for two separate networks that vary in size by a factor of100. Third, the change in the clearness index correlation coefficientconverges to the clearness correlation coefficient as the time intervalincreases. This convergence is predicted based on the form of theempirical model because ΔClearnessPower approaches one as Δt becomeslarge.

Equation (46) and (47) were validated by calculating the averagevariance of the clearness index and the variance of the change in theclearness index across the 25 locations in each network for everyhalf-hour time period. FIGS. 15A-15F are graphs depicting, by way ofexample, the predicted versus the measured variances of clearnessindexes using different reference time intervals. FIGS. 16A-16F aregraphs depicting, by way of example, the predicted versus the measuredvariances of change in clearness indexes using different reference timeintervals. FIGS. 15A-15F and 16A-16F suggest that the predicted resultsare similar to the measured results.

Discussion

The point-to-point correlation coefficients calculated using theempirical forms described supra refer to the locations of specificphotovoltaic power production sites. Importantly, note that the dataused to calculate these coefficients was not obtained from time sequencemeasurements taken at the points themselves. Rather, the coefficientswere calculated from fleet-level data (cloud speed), fixed fleet data(distances between points), and user-specified data (time interval).

The empirical relationships of the foregoing types of empiricalrelationships may be used to rapidly compute the coefficients that arethen used in the fundamental mathematical relationships. The methodologydoes not require that these specific empirical models be used andimproved models will become available in the future with additional dataand analysis.

Example

This section provides a complete illustration of how to apply themethodology using data from the Napa network of 25 irradiance sensors onNov. 21, 2010. In this example, the sensors served as proxies for anactual 1 kW photovoltaic fleet spread evenly over the geographicalregion as defined by the sensors. For comparison purposes, a directmeasurement approach is used to determine the power of this fleet andthe change in power, which is accomplished by adding up the 10-secondoutput from each of the sensors and normalizing the output to a 1 kWsystem. FIGS. 17A-17F are graphs and a diagram depicting, by way ofexample, application of the methodology described herein to the Napanetwork.

The predicted behavior of the hypothetical photovoltaic fleet wasseparately estimated using the steps of the methodology described supra.The irradiance data was measured using ground-based sensors, althoughother sources of data could be used, including from existingphotovoltaic systems or satellite imagery. As shown in FIG. 17A, thedata was collected on a day with highly variable clouds with one-minuteglobal horizontal irradiance data collected at one of the 25 locationsfor the Napa network and specific 10-second measured power outputrepresented by a blue line. This irradiance data was then converted fromglobal horizontal irradiance to a clearness index. The mean clearnessindex, variance of clearness index, and variance of the change inclearness index were then calculated for every 15-minute period in theday. These calculations were performed for each of the 25 locations inthe network. Satellite-based data or a statistically-significant subsetof the ground measurement locations could have also served in place ofthe ground-based irradiance data. However, if the data had beencollected from satellite regions, an additional translation from areastatistics to average point statistics would have been required. Theaveraged irradiance statistics from Equations (1), (10), and (11) areshown in FIG. 17B, where standard deviation (σ) is presented, instead ofvariance (σ²) to plot each of these values in the same units.

In this example, the irradiance statistics need to be translated sincethe data were recorded at a time interval of 60 seconds, but the desiredresults are at a 10-second resolution. The translation was performedusing Equations (46) and (47) and the result is presented in FIG. 17C.

The details of the photovoltaic fleet configuration were then obtained.The layout of the fleet is presented in FIG. 17D. The details includethe location of the each photovoltaic system (latitude and longitude),photovoltaic system rating ( 1/25 kW), and system orientation (all arehorizontal).

Equation (43), and its associated component equations, were used togenerate the time series data for the photovoltaic fleet with theadditional specification of the specific empirical models, as describedin Equations (44) through (47). The resulting fleet power and change inpower is presented represented by the red lines in FIGS. 16E and 16F.

Probability Density Function

The conversion from area statistics to point statistics relied upon twoterms A_(Kt) and A_(ΔKt) to calculate σ_(Kt) ² and aσ_(ΔKt) ²,respectively. This section considers these terms in more detail. Forsimplicity, the methodology supra applies to both Kt and ΔKt, so thisnotation is dropped. Understand that the correlation coefficient ρ^(ij)could refer to either the correlation coefficient for clearness index orthe correlation coefficient for the change in clearness index, dependingupon context. Thus, the problem at hand is to evaluate the followingrelationship:

$\begin{matrix}{A = {\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\rho^{i,j}}}}} & (48)\end{matrix}$

The computational effort required to calculate the correlationcoefficient matrix can be substantial. For example, suppose that the onewants to evaluate variance of the sum of points within a one-squarekilometer satellite region by breaking the region into one millionsquare meters (1,000 meters by 1,000 meters). The complete calculationof this matrix requires the examination of one trillion (10¹²) locationpair combinations. A heuristical approach to simplifying the solutionspace from exponential to linear is discussed supra with reference toFIGS. 7-10.

Discrete Formulation

The calculation can be simplified using the observation that many of theterms in the correlation coefficient matrix are identical. For example,the covariance between any of the one million points and themselvesis 1. This observation can be used to show that, in the case of arectangular region that has dimension of H by W points (total of N) andthe capacity is equal distributed across all parts of the region that:

$\begin{matrix}{{\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\rho^{i,j}}}} = {\left( \frac{1}{N^{2}} \right)\left\lbrack {{\sum\limits_{i = 0}^{H - 1}{\sum\limits_{j = 0}^{i}{{2^{k}\left\lbrack {\left( {H - i} \right)\left( {W - j} \right)} \right\rbrack}\rho^{d}}}} + {\sum\limits_{i = 0}^{W - 1}{\sum\limits_{j = 0}^{i}{2^{k}\left\lbrack {\left( {W - i} \right)\left( {H - j} \right)\text{?}\text{?}\text{indicates text missing or illegible when filed}} \right.}}}} \right.}} & (49)\end{matrix}$

where:

−1, when i=0 and j=0

k=1, when j=0 or j=i

2, when 0<j<i

When the region is a square, a further simplification can be made.

$\begin{matrix}{{\left( \frac{1}{N^{2}} \right){\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}\rho^{i,j}}}} = {\left( \frac{1}{N^{2}} \right)\left\lbrack {\sum\limits_{i = 0}^{\sqrt{N} - 1}{\sum\limits_{j = 0}^{i}{2^{k}\left( {\sqrt{N} - i} \right)\left( {\sqrt{N} - j} \right)\rho^{d}}}} \right\rbrack}} & (50)\end{matrix}$

where:

0, when i=0 and j=0

k=2, when j=0 or j=i, and

3, when 0<j<i

$d = {\left( \sqrt{i^{2} + j^{2}} \right){\left( \frac{\sqrt{Area}}{\sqrt{N} - 1} \right).}}$

The benefit of Equation (50) is that there are

$\frac{N - \sqrt{N}}{2}$

rather than N² unique combinations that need to be evaluated. In theexample above, rather than requiring one trillion possible combinations,the calculation is reduced to one-half million possible combinations.

Continuous Formulation

Even given this simplification, however, the problem is stillcomputationally daunting, especially if the computation needs to beperformed repeatedly in the time series. Therefore, the problem can berestated as a continuous formulation in which case a proposedcorrelation function may be used to simplify the calculation. The onlyvariable that changes in the correlation coefficient between any of thelocation pairs is the distance between the two locations; all othervariables are the same for a given calculation. As a result, Equation(50) can be interpreted as the combination of two factors: theprobability density function for a given distance occurring and thecorrelation coefficient at the specific distance.

Consider the probability density function. The actual probability of agiven distance between two pairs occurring was calculated for a 1,000meter×1,000 meter grid in one square meter increments. The evaluation ofone trillion location pair combination possibilities was evaluated usingEquation (48) and by eliminating the correlation coefficient from theequation. FIG. 18 is a graph depicting, by way of example, an actualprobability distribution for a given distance between two pairs oflocations, as calculated for a 1,000 meter×1,000 meter grid in onesquare meter increments.

The probability distribution suggests that a continuous approach can betaken, where the goal is to find the probability density function basedon the distance, such that the integral of the probability densityfunction times the correlation coefficient function equals:

A=∫f(D)ρ(d)dD   (51)

An analysis of the shape of the curve shown in FIG. 18 suggests that thedistribution can be approximated through the use of two probabilitydensity functions. The first probability density function is a quadraticfunction that is valid between 0 and √{square root over (Area)}.

$\begin{matrix}{f_{Quad} = \left\{ \begin{matrix}{\left( \frac{6}{Area} \right)\left( {D - \frac{D^{2}}{\sqrt{Area}}} \right)} & {{{for}\mspace{14mu} 0} \leq D \leq \sqrt{Area}} \\0 & {{{for}\mspace{14mu} D} > \sqrt{Area}}\end{matrix} \right.} & (52)\end{matrix}$

This function is a probability density function because integratingbetween 0 and √{square root over (Area)} equals 1 (i.e., P[0≤D≤√{squareroot over (Area)}]=∫₀ ^(√{square root over (Area)})

dD=1).

The second function is a normal distribution with a mean of Area andstandard deviation of 0.1√{square root over (Area)}.

$\begin{matrix}{f_{Norm} = {\left( \frac{1}{{0.1}*\sqrt{Area}} \right)\left( \frac{1}{\sqrt{2\pi}} \right)e^{{- {(\frac{1}{2})}}{(\frac{D - \sqrt{Area}}{{0.1}*\sqrt{Area}})}^{2}}}} & (53)\end{matrix}$

Likewise, integrating across all values equals 1.

To construct the desired probability density function, take, forinstance, 94 percent of the quadratic density function plus 6 of thenormal density function.

f=0.94∫₀ ^(√{square root over (Area)})

dD+0.06∫_(−∞) ^(+√) f _(Norm) dD   (54)

FIG. 19 is a graph depicting, by way of example, a matching of theresulting model to an actual distribution.

The result is that the correlation matrix of a square area with uniformpoint distribution as N gets large can be expressed as follows, firstdropping the subscript on the variance since this equation will work forboth Kt and ΔKt.

A≈[0.94∫₀ ^(√{square root over (Area)})

ρ(D)dD+0.06∫_(−∞) ^(+∞)f_(Norm)ρ(D

  (55)

where ρ(D) is a function that expresses the correlation coefficient as afunction of distance (D).

Area to Point Conversion Using Exponential Correlation Coefficient

Equation (55) simplifies the problem of calculating the correlationcoefficient and can be implemented numerically once the correlationcoefficient function is known. This section demonstrates how a closedform solution can be provided, if the functional form of the correlationcoefficient function is exponential.

Noting the empirical results as shown in the graph in FIGS. 13A-13F, anexponentially decaying function can be taken as a suitable form for thecorrelation coefficient function. Assume that the functional form ofcorrelation coefficient function equals:

$\begin{matrix}{{\rho (D)} = e^{\frac{xD}{\sqrt{Area}}}} & (56)\end{matrix}$

Let

be the solution to ƒ₀ ^(√{square root over (Area)})

ρ(D)dD.

$\begin{matrix}{{Quad} = {{\int_{0}^{\sqrt{Area}}{f_{Quad}{\rho (D)}dD}} = {\left( \frac{6}{Area} \right){\int_{0}^{\sqrt{Area}}{{\left( {D - \frac{D^{2}}{\sqrt{Area}}} \right)\left\lbrack e^{\frac{xD}{\sqrt{Area}}} \right\rbrack}{dD}}}}}} & (57)\end{matrix}$

Integrate to solve.

$\begin{matrix}{{Quad} = {(6)\left\lbrack {{\left( {{\frac{X}{\sqrt{Area}}D} - 1} \right)e^{\frac{xD}{\sqrt{Area}}}} - {\left( {{\left( \frac{X}{\sqrt{Area}} \right)^{2}D^{2}} - {2\frac{X}{\sqrt{Area}}D} + 2} \right)e^{\frac{xD}{\sqrt{Area}}}}} \right\rbrack}} & (58)\end{matrix}$

Complete the result by evaluating at D equal to √{square root over(Area)} for the upper bound and 0 for the lower bound. The result is:

$\begin{matrix}{{Quad} = {\left( \frac{6}{x^{3}} \right)\left\lbrack {{\left( {x - 2} \right)\left( {e^{x} + 1} \right)} + 4} \right\rbrack}} & (59)\end{matrix}$

Next, consider the solution to ∫_(−∞) ^(+∞)f_(Norm)ρ^((D)dD), which willbe called Norm.

$\begin{matrix}{{Norm} = {\left( \frac{1}{\sigma} \right)\left( \frac{1}{\sqrt{2\pi}} \right){\int_{- \infty}^{+ \infty}{e^{{- {(\frac{1}{2})}}{(\frac{D - \mu}{\sigma})}^{2}}e^{\frac{xD}{\sqrt{Area}}}{dD}}}}} & (60)\end{matrix}$

Where μ=√{square root over (Area)} and σ=0.1√{square root over (Area)}.Simplifying:

$\begin{matrix}{{Norm} = {\left\lbrack e^{\frac{x}{\sqrt{Area}}{({\mu + {{({\frac{1}{2}\frac{x}{\sqrt{Area}}})}\sigma^{2}}})}} \right\rbrack {\left( \frac{1}{\sigma} \right) \cdot \left( \frac{1}{\sqrt{2\pi}} \right)}{\int_{- \infty}^{+ \infty}e^{- {{(\frac{1}{2})}\lbrack\frac{D - {({\mu + {\frac{x}{\sqrt{Area}}\sigma^{2}}})}}{\sigma}\rbrack}_{dD}^{2}}}}} & (61)\end{matrix}$

Substitute

$z = \frac{D - \left( {\mu + {\frac{x}{\sqrt{Area}}\sigma^{2}}} \right)}{\sigma}$

and σdz=dD .

$\begin{matrix}{{Norm} = {\left\lbrack e^{\frac{x}{\sqrt{Area}}{({\mu + {{({\frac{1}{2}\frac{x}{\sqrt{Area}}})}\sigma^{2}}})}} \right\rbrack \left( \frac{1}{\sqrt{2\pi}} \right){\int_{- \infty}^{+ \infty}{e^{{- {(\frac{1}{2})}}z^{2}}{dz}}}}} & (62)\end{matrix}$

Integrate and solve.

$\begin{matrix}{{Norm} = e^{\frac{x}{\sqrt{Area}}{({\mu + {{({\frac{1}{2}\frac{x}{\sqrt{Area}}})}\sigma^{2}}})}}} & (63)\end{matrix}$

Substitute the mean of √{square root over (Area)} and the standarddeviation of 0.1√{square root over (Area)} into Equation (63).

Norm=e ^(x(1+0.005x))   (64)

Substitute the solutions for Quad and Norm back into Equation (55). Theresult is the ratio of the area variance to the average point variance.This ratio was referred to as A (with the appropriate subscripts andsuperscripts) supra.

$\begin{matrix}{A = {{{0.9}4{\left( \frac{6}{x^{3}} \right)\left\lbrack {{\left( {x - 2} \right)\left( {e^{x} + 1} \right)} + 4} \right\rbrack}} + {0.06e^{x{({1 + {0.005x}})}}}}} & (65)\end{matrix}$

Example

This section illustrates how to calculate A for the clearness index fora satellite pixel that covers a geographical surface area of 1 km by 1km (total area of 1,000,000 m²), using a 60-second time interval, and 6meter per second cloud speed. Equation (56) required that thecorrelation coefficient be of the form

$e^{\frac{xD}{\sqrt{Area}}}.$

The empirically derived result in Equation (44) can be rearranged andthe appropriate substitutions made to show that the correlationcoefficient of the clearness index equals

${\exp \left\lbrack \frac{\left( {{\ln \; \Delta \; t} - 9.3} \right)D}{1000\mspace{14mu} {CloudSpeed}} \right\rbrack}.$

Multiply the exponent by

$\frac{\sqrt{Area}}{\sqrt{Area}},$

so that the correlation coefficient equals

$\exp {\left\{ {\left\lbrack \frac{\left( {{\ln \; \Delta \; t} - 9.3} \right)\sqrt{Area}}{1000\mspace{14mu} {CloudSpeed}} \right\rbrack \left\lbrack \frac{D}{\sqrt{Area}} \right\rbrack} \right\}.}$

This expression is now in the correct form to apply Equation (65), where

$x = {\frac{\left( {{\ln \; \Delta \; t} - 9.3} \right)\sqrt{Area}}{1000\mspace{14mu} {CloudSpeed}}.}$

Inserting the assumptions results in

${x = {\frac{\left( {{\ln \; 60} - 9.3} \right)\sqrt{1,000,000}}{1000 \times 6} = {- 0.86761}}},$

which is applied to Equation (65). The result is that A equals 65percent, that is, the variance of the clearness index of the satellitedata collected over a 1 km² region corresponds to 65 percent of thevariance when measured at a specific point. A similar approach can beused to show that the A equals 27 percent for the change in clearnessindex. FIG. 20 is a graph depicting, by way of example, resultsgenerated by application of Equation (65).

Time Lag Correlation Coefficient

This section presents an alternative approach to deriving the time lagcorrelation coefficient. The variance of the sum of the change in theclearness index equals:

σ_(ΣΔKt) ²=VAR[Σ(Kt ^(Δt) −Kt)]  (66)

where the summation is over N locations. This value and thecorresponding subscripts have been excluded for purposes of notationalsimplicity.

Divide the summation into two parts and add several constants to theequation:

$\begin{matrix}{\sigma_{\sum{\Delta \; {Kt}}}^{2} = {{VAR}\left\lbrack {{\sigma_{\sum{Kt}^{\Delta \; t}}\left( \frac{\sum{Kt}^{\Delta \; t}}{\sigma_{\sum{Kt}^{\Delta \; t}}} \right)} - {\sigma_{\sum{Kt}}\left( \frac{\sum{Kt}}{\sigma_{\sum{Kt}}} \right)}} \right\rbrack}} & (67)\end{matrix}$

Since σ_(ΣKt) ^(Δt)≈σ_(ΣKt) (or σ_(ΣKt) ^(Δt)=σ_(ΣKt) if the first termin Kt and the last term in Kt^(Δt) are the same):

$\begin{matrix}{\sigma_{\sum{\Delta \; {Kt}}}^{2} = {\sigma_{\sum{Kt}}^{2}{{VAR}\left\lbrack {\frac{\sum{Kt}^{\Delta \; t}}{\sigma_{\sum{Kt}^{\Delta \; t}}} - \frac{\sum{Kt}}{\sigma_{\sum{Kt}}}} \right\rbrack}}} & (68)\end{matrix}$

The variance term can be expanded as follows:

$\begin{matrix}{\sigma_{\sum{\Delta \; {Kt}}}^{2} = {\sigma_{\sum{Kt}}^{2}\left\{ {\frac{{VAR}\left\lbrack {\sum{Kt}^{\Delta \; t}} \right\rbrack}{\sigma_{\sum{Kt}^{\Delta \; t}}^{2}} + \frac{{VAR}\left\lbrack {\sum{Kt}} \right\rbrack}{\sigma_{\sum{Kt}}^{2}} - \frac{2\; {{COV}\left\lbrack {{\sum{Kt}},{\sum{Kt}^{\Delta \; t}}} \right\rbrack}}{\sigma_{\sum{Kt}}\sigma_{\sum{Kt}^{\Delta \; t}}}} \right\}}} & (69)\end{matrix}$

Since COV[ΣKt,ΣKt^(Δt)]=σ_(ΣKt)σ_(ΣKt) ^(Δt)ρ^(ΣKt,ΣKt) ^(Δt) , thefirst two terms equal one and the covariance term is replaced by thecorrelation coefficient.

σ_(ΣΔKt) ²=2σ_(ΣKt) ²(1−ρ^(ΣKt,ΣKt) ^(Δt) )   (70)

This expression rearranges to:

$\begin{matrix}{\rho^{{\sum{Kt}},{\sum{Kt}^{\Delta \; t}}} = {1 - \left( {\frac{1}{2}\frac{\sigma_{\Delta \; {Kt}}^{2}}{\sigma_{Kt}^{2}}} \right)}} & (71)\end{matrix}$

Assume that all photovoltaic plant ratings, orientations, and areaadjustments equal to one, calculate statistics for the clearness aloneusing the equations described supra and then substitute. The result is:

$\begin{matrix}{\rho^{{\sum{Kt}},{\sum{Kt}^{\Delta \; t}}} = {1 - \frac{P^{\Delta \; {Kt}}\sigma \frac{2}{\Delta \; {Kt}}}{2\; P^{Kt}\sigma \; \frac{2}{Kt}}}} & (72)\end{matrix}$

Relationship Between Time Lag Correlation Coefficient and Power/Changein Power Correlation Coefficient

This section derives the relationship between the time lag correlationcoefficient and the correlation between the series and the change in theseries for a single location.

$\rho^{P,{\Delta \; P}} = {\frac{{COV}\left\lbrack {P,{\Delta \; P}} \right\rbrack}{\sqrt{\sigma_{P}^{2}\sigma_{\Delta \; P}^{2}}} = {\frac{{COV}\left\lbrack {P,{P^{\Delta \; t} - P}} \right\rbrack}{\sqrt{\sigma_{P}^{2}\sigma_{\Delta \; P}^{2}}} = \frac{{{COV}\left\lbrack {P,P^{\Delta \; t}} \right\rbrack} - \sigma_{P}^{2}}{\sqrt{\sigma_{P}^{2}\sigma_{\Delta \; P}^{2}}}}}$

Since σ_(ΔP) ²=VAR[P^(Δt)−P]=σ_(P) ²+σ_(P) _(Δt) ²−2COV[P, P^(Δt)], and

COV[P,P^(Δt)]=ρ^(P,P) ^(Δt) √{square root over (σ_(P) ²σ_(P) _(Δt) ²)},then:

$\rho^{P,{\Delta \; P}} = \frac{{\rho^{P,P^{\Delta \; t}}\sqrt{\sigma_{P}^{2}\sigma_{P^{\Delta \; t}}^{2}}} - \sigma_{P}^{2}}{\sqrt{\sigma_{P}^{2}\left( {\sigma_{P}^{2} + \sigma_{P^{\Delta \; t}}^{2} - {2\; \rho^{P,P^{\Delta \; t}}\sqrt{\sigma_{P}^{2}\sigma_{P^{\Delta \; t}}^{2}}}} \right)}}$

Since σ_(P) ²≈σ_(P) _(Δt) ², this expression can be further simplified.Then, square both expression and solve for the time lag correlationcoefficient:

ρ^(P,P) ^(Δt) =1−2(ρ^(P,ΔP))²

Correlation Coefficients Between Two Regions

Assume that the two regions are squares of the same size, each side withN points, that is, a matrix with dimensions of √{square root over (N)}by √{square root over (N)} points, where √{square root over (N)} is aninteger, but are separated by one or more regions. Thus:

$\begin{matrix}{{{\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{N}{\left( \frac{1}{N^{2}} \right)\rho^{i,j}}}} = {\left( \frac{1}{N^{2}} \right)\left\lbrack {\sum\limits_{i = 0}^{\sqrt{N} - 1}{\sum\limits_{j = {1 - \sqrt{N}}}^{\sqrt{N} - 1}{{k\left( {\sqrt{N} - i} \right)}\left( {\sqrt{N} - {j}} \right)\rho^{d}}}} \right\rbrack}}\mspace{79mu} {{where}\text{:}}\mspace{79mu} {{k = \begin{matrix}{1,} & {{{when}\mspace{14mu} i} = 0} \\{2,} & {{{when}\mspace{14mu} i} > 0}\end{matrix}},{and}}\text{}\mspace{79mu} {{d = {\left( \sqrt{i^{2} + \left( {j + {M\sqrt{N}}} \right)^{2}} \right)\left( \frac{\sqrt{Area}}{\sqrt{N} - 1} \right)}},}} & (73)\end{matrix}$

and

such that M equals the number of regions.

FIG. 21 is a graph depicting, by way of example, the probability densityfunction when regions are spaced by zero to five regions. FIG. 21suggests that the probability density function can be estimated usingthe following distribution:

$\begin{matrix}{f = \left\{ \begin{matrix}{1 - \left( \frac{{Spacing} - D}{\sqrt{Area}} \right)} & {{{{for}\mspace{14mu} {Spacing}} - \sqrt{Area}} \leq D \leq {Spacing}} \\{1 + \left( \frac{{Spacing} - D}{\sqrt{Area}} \right)} & {{{for}\mspace{14mu} {Spacing}} \leq D \leq {{Spacing} + \sqrt{Area}}} \\0 & {{all}\mspace{14mu} {else}}\end{matrix} \right.} & (74)\end{matrix}$

This function is a probability density function because the integrationover all possible values equals zero. FIG. 22 is a graph depicting, byway of example, results by application of this model.

While the invention has been particularly shown and described asreferenced to the embodiments thereof, those skilled in the art willunderstand that the foregoing and other changes in form and detail maybe made therein without departing from the spirit and scope.

What is claimed is:
 1. A system for variance-based photovoltaic fleetpower statistics building with the aid of a digital computer,comprising: a storage comprising a set of pixels in satellite imagerydata of overhead sky clearness that have been correlated to a boundedarea within a geographic region, each pixel representing collectiveirradiance over a plurality of points within the bounded area, eachpoint suitable for operation of a photovoltaic system comprised in aphotovoltaic fleet, the photovoltaic fleet connected to a power grid;and a computer coupled to the storage and comprising a processorconfigured to execute code in a memory, the code comprising: adetermination module configured to determine an area clearness indexbased on the collective irradiances, as represented by the set of pixelscorrelated to the bounded area; a definition module configured to set acondition for ending a determination of a variance of the area clearnessindex; an evaluation module configured to perform the determination ofthe variance of the area clearness index until the condition is met, theevaluation module comprising: a point module configured to choose apoint within the bounded area; a selection module configured to selectat least some of the points within the bounded area that have notalready been evaluated for pairing with the chosen point; a pairingmodule configured to pair the chosen point with each of the selectedpoints; and a calculation module configured to calculate a covariancefor at least some of the point pairings and to add the calculatedcovariance for at least some of the pairs to the variance of the areaclearness index; and a power statistics module configured to build apower statistics for the photovoltaic fleet using the variance of thearea clearness index, wherein the power grid is operated based on thepower statistics.
 2. A system according to claim 1, wherein thecondition comprises all of the points having been processed.
 3. A systemaccording to claim 1, further comprising: a coefficient moduleconfigured to find a correlation coefficient for each of the pointpairings; and a value module configured to determine a zero correlationvalue for each chosen point, wherein the covariance for one of the pointpairings is calculated and added to the variance only if the correlationcoefficient for that point pairing exceeds the zero correlation value.4. A system according to claim 1, further comprising: a distance moduleconfigured to determine a zero correlation distance for the chosenpoint, wherein the selected points are within the zero correlationdistance of the chosen point.
 5. A system according to claim 4, whereina set of the points located within the zero correlation distance fromthe chosen point is represented as one of a grid centered on the chosenpoint or a correlation region centered on the chosen point.
 6. A systemaccording to claim 4, wherein a set of the points located within thezero correlation distance from the chosen point is represented as acorrelation region centered on the chosen point, a shape of thecorrelation region comprising at least one of a rectangle, a square, anda circle.
 7. A system according to claim 6, wherein the correlationregion is defined using latitude and longitude.
 8. A system according toclaim 4, wherein the correlation distance is determined in meters,further comprising: a conversion module configured to convert thecorrelation distance into degrees and determine a latitude and longitudeof the chosen point.
 9. A system according to claim 4, furthercomprising: a function module configured to determine the zerocorrelation distance as a function of a distance from the chosen point,cloud speed, and a time interval relating to a time resolution ofobservation for the collective irradiance.
 10. A system according toclaim 9, wherein the cloud speed varies across the points and the zerocorrelation distance is calculated for each of the points.
 11. A methodfor variance-based photovoltaic fleet power statistics building with theaid of a digital computer, comprising: maintaining in a storage a set ofpixels in satellite imagery data of overhead sky clearness that havebeen correlated to a bounded area within a geographic region, each pixelrepresenting collective irradiance over a plurality of points within thebounded area, each point suitable for operation of a photovoltaic systemcomprised in a photovoltaic fleet, the photovoltaic fleet connected to apower grid; determining by a computer, the computer comprising aprocessor and coupled to the storage, an area clearness index based onthe collective irradiances, as represented by the set of pixelscorrelated to the bounded area; setting by the computer a condition forending a determination of a variance of the area clearness index;performing by the computer the determination of the variance of the areaclearness index until the condition is met, comprising: choosing a pointwithin the bounded area; selecting at least some of the points withinthe bounded area that have not already been evaluated for pairing withthe chosen point; pairing the chosen point with each of the selectedpoints; and calculating a covariance for at least some of the pointpairings and adding the calculated covariance for at least some of thepairs to the variance of the area clearness index; and building by thecomputer a power statistics for the photovoltaic fleet using thevariance of the area clearness index, wherein the power grid is operatedbased on the power statistics.
 12. A method according to claim 11,wherein the condition comprises all of the points having been processed.13. A method according to claim 11, further comprising: finding acorrelation coefficient for each of the point pairings; and determininga zero correlation value for each chosen point, wherein the covariancefor one of the point pairings is calculated and added to the varianceonly if the correlation coefficient for that point pairing exceeds thezero correlation value.
 14. A method according to claim 11, furthercomprising: determining a zero correlation distance for the chosenpoint, wherein the selected points are within the zero correlationdistance of the chosen point.
 15. A method according to claim 14,wherein a set of the points located within the zero correlation distancefrom the chosen point is represented as one of a grid centered on thechosen point or a correlation region centered on the chosen point.
 16. Amethod according to claim 14, wherein a set of the points located withinthe zero correlation distance from the chosen point is represented as acorrelation region centered on the chosen point, a shape of thecorrelation region comprising at least one of a rectangle, a square, anda circle.
 17. A method according to claim 16, wherein the correlationregion is defined using latitude and longitude.
 18. A method accordingto claim 14, wherein the correlation distance is determined in meters,further comprising: converting the correlation distance into degrees anddetermining a latitude and longitude of the chosen point.
 19. A methodaccording to claim 14, further comprising: determining the zerocorrelation distance as a function of a distance from the chosen point,cloud speed, and a time interval relating to a time resolution ofobservation for the collective irradiance.
 20. A method according toclaim 19, wherein the cloud speed varies across the points and the zerocorrelation distance is calculated for each of the points.